Daily Papers

Language is typically modelled with discrete sequences. However, the most successful approaches to language modelling, namely neural networks, are continuous and smooth function approximators. In this work, we show that Transformer-based language models implicitly learn to represent sentences as continuous-time functions defined over a continuous input space. This phenomenon occurs in most state-of-the-art Large Language Models (LLMs), including Llama2, Llama3, Phi3, Gemma, Gemma2, and Mistral, and suggests that LLMs reason about language in ways that fundamentally differ from humans. Our work formally extends Transformers to capture the nuances of time and space continuity in both input and output space. Our results challenge the traditional interpretation of how LLMs understand language, with several linguistic and engineering implications.

This paper introduces Bayesian Flow Networks (BFNs), a new class of generative model in which the parameters of a set of independent distributions are modified with Bayesian inference in the light of noisy data samples, then passed as input to a neural network that outputs a second, interdependent distribution. Starting from a simple prior and iteratively updating the two distributions yields a generative procedure similar to the reverse process of diffusion models; however it is conceptually simpler in that no forward process is required. Discrete and continuous-time loss functions are derived for continuous, discretised and discrete data, along with sample generation procedures. Notably, the network inputs for discrete data lie on the probability simplex, and are therefore natively differentiable, paving the way for gradient-based sample guidance and few-step generation in discrete domains such as language modelling. The loss function directly optimises data compression and places no restrictions on the network architecture. In our experiments BFNs achieve competitive log-likelihoods for image modelling on dynamically binarized MNIST and CIFAR-10, and outperform all known discrete diffusion models on the text8 character-level language modelling task.

We introduce Functional Diffusion Processes (FDPs), which generalize score-based diffusion models to infinite-dimensional function spaces. FDPs require a new mathematical framework to describe the forward and backward dynamics, and several extensions to derive practical training objectives. These include infinite-dimensional versions of Girsanov theorem, in order to be able to compute an ELBO, and of the sampling theorem, in order to guarantee that functional evaluations in a countable set of points are equivalent to infinite-dimensional functions. We use FDPs to build a new breed of generative models in function spaces, which do not require specialized network architectures, and that can work with any kind of continuous data. Our results on real data show that FDPs achieve high-quality image generation, using a simple MLP architecture with orders of magnitude fewer parameters than existing diffusion models.

We propose a unified framework to study policy evaluation (PE) and the associated temporal difference (TD) methods for reinforcement learning in continuous time and space. We show that PE is equivalent to maintaining the martingale condition of a process. From this perspective, we find that the mean--square TD error approximates the quadratic variation of the martingale and thus is not a suitable objective for PE. We present two methods to use the martingale characterization for designing PE algorithms. The first one minimizes a "martingale loss function", whose solution is proved to be the best approximation of the true value function in the mean--square sense. This method interprets the classical gradient Monte-Carlo algorithm. The second method is based on a system of equations called the "martingale orthogonality conditions" with test functions. Solving these equations in different ways recovers various classical TD algorithms, such as TD(lambda), LSTD, and GTD. Different choices of test functions determine in what sense the resulting solutions approximate the true value function. Moreover, we prove that any convergent time-discretized algorithm converges to its continuous-time counterpart as the mesh size goes to zero, and we provide the convergence rate. We demonstrate the theoretical results and corresponding algorithms with numerical experiments and applications.

Integral reinforcement learning (IntRL) demands the precise computation of the utility function's integral at its policy evaluation (PEV) stage. This is achieved through quadrature rules, which are weighted sums of utility functions evaluated from state samples obtained in discrete time. Our research reveals a critical yet underexplored phenomenon: the choice of the computational method -- in this case, the quadrature rule -- can significantly impact control performance. This impact is traced back to the fact that computational errors introduced in the PEV stage can affect the policy iteration's convergence behavior, which in turn affects the learned controller. To elucidate how computation impacts control, we draw a parallel between IntRL's policy iteration and Newton's method applied to the Hamilton-Jacobi-Bellman equation. In this light, computational error in PEV manifests as an extra error term in each iteration of Newton's method, with its upper bound proportional to the computational error. Further, we demonstrate that when the utility function resides in a reproducing kernel Hilbert space (RKHS), the optimal quadrature is achievable by employing Bayesian quadrature with the RKHS-inducing kernel function. We prove that the local convergence rates for IntRL using the trapezoidal rule and Bayesian quadrature with a Mat\'ern kernel to be O(N^{-2}) and O(N^{-b}), where N is the number of evenly-spaced samples and b is the Mat\'ern kernel's smoothness parameter. These theoretical findings are finally validated by two canonical control tasks.

Effective data-driven PDE forecasting methods often rely on fixed spatial and / or temporal discretizations. This raises limitations in real-world applications like weather prediction where flexible extrapolation at arbitrary spatiotemporal locations is required. We address this problem by introducing a new data-driven approach, DINo, that models a PDE's flow with continuous-time dynamics of spatially continuous functions. This is achieved by embedding spatial observations independently of their discretization via Implicit Neural Representations in a small latent space temporally driven by a learned ODE. This separate and flexible treatment of time and space makes DINo the first data-driven model to combine the following advantages. It extrapolates at arbitrary spatial and temporal locations; it can learn from sparse irregular grids or manifolds; at test time, it generalizes to new grids or resolutions. DINo outperforms alternative neural PDE forecasters in a variety of challenging generalization scenarios on representative PDE systems.

This paper aims to extend the Structured Knowledge Accumulation (SKA) framework recently proposed by mahi2025ska. We introduce two core concepts: the Tensor Net function and the characteristic time property of neural learning. First, we reinterpret the learning rate as a time step in a continuous system. This transforms neural learning from discrete optimization into continuous-time evolution. We show that learning dynamics remain consistent when the product of learning rate and iteration steps stays constant. This reveals a time-invariant behavior and identifies an intrinsic timescale of the network. Second, we define the Tensor Net function as a measure that captures the relationship between decision probabilities, entropy gradients, and knowledge change. Additionally, we define its zero-crossing as the equilibrium state between decision probabilities and entropy gradients. We show that the convergence of entropy and knowledge flow provides a natural stopping condition, replacing arbitrary thresholds with an information-theoretic criterion. We also establish that SKA dynamics satisfy a variational principle based on the Euler-Lagrange equation. These findings extend SKA into a continuous and self-organizing learning model. The framework links computational learning with physical systems that evolve by natural laws. By understanding learning as a time-based process, we open new directions for building efficient, robust, and biologically-inspired AI systems.

We introduce Temporal Variational Implicit Neural Representations (TV-INRs), a probabilistic framework for modeling irregular multivariate time series that enables efficient individualized imputation and forecasting. By integrating implicit neural representations with latent variable models, TV-INRs learn distributions over time-continuous generator functions conditioned on signal-specific covariates. Unlike existing approaches that require extensive training, fine-tuning or meta-learning, our method achieves accurate individualized predictions through a single forward pass. Our experiments demonstrate that with a single TV-INRs instance, we can accurately solve diverse imputation and forecasting tasks, offering a computationally efficient and scalable solution for real-world applications. TV-INRs excel especially in low-data regimes, where it outperforms existing methods by an order of magnitude in mean squared error for imputation task.

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

This work addresses continuous space-time video super-resolution (C-STVSR) that aims to up-scale an input video both spatially and temporally by any scaling factors. One key challenge of C-STVSR is to propagate information temporally among the input video frames. To this end, we introduce a space-time local implicit neural function. It has the striking feature of learning forward motion for a continuum of pixels. We motivate the use of forward motion from the perspective of learning individual motion trajectories, as opposed to learning a mixture of motion trajectories with backward motion. To ease motion interpolation, we encode sparsely sampled forward motion extracted from the input video as the contextual input. Along with a reliability-aware splatting and decoding scheme, our framework, termed MoTIF, achieves the state-of-the-art performance on C-STVSR. The source code of MoTIF is available at https://github.com/sichun233746/MoTIF.

Verifiable delay functions have found a lot of applications in blockchain technology in recent times. Continuous verifiable delay functions are an improvement over the basic notion of VDFs with recursive capabilities. We are proposing the application of VDF for constructing more space time-efficient provers and simulators required for the iterative non-interactive zero-knowledge systems.

Temporal data such as time series can be viewed as discretized measurements of the underlying function. To build a generative model for such data we have to model the stochastic process that governs it. We propose a solution by defining the denoising diffusion model in the function space which also allows us to naturally handle irregularly-sampled observations. The forward process gradually adds noise to functions, preserving their continuity, while the learned reverse process removes the noise and returns functions as new samples. To this end, we define suitable noise sources and introduce novel denoising and score-matching models. We show how our method can be used for multivariate probabilistic forecasting and imputation, and how our model can be interpreted as a neural process.

Recent years have seen significant progress in developing spiking neural networks (SNNs) as a potential solution to the energy challenges posed by conventional artificial neural networks (ANNs). However, our theoretical understanding of SNNs remains relatively limited compared to the ever-growing body of literature on ANNs. In this paper, we study a discrete-time model of SNNs based on leaky integrate-and-fire (LIF) neurons, referred to as discrete-time LIF-SNNs, a widely used framework that still lacks solid theoretical foundations. We demonstrate that discrete-time LIF-SNNs with static inputs and outputs realize piecewise constant functions defined on polyhedral regions, and more importantly, we quantify the network size required to approximate continuous functions. Moreover, we investigate the impact of latency (number of time steps) and depth (number of layers) on the complexity of the input space partitioning induced by discrete-time LIF-SNNs. Our analysis highlights the importance of latency and contrasts these networks with ANNs employing piecewise linear activation functions. Finally, we present numerical experiments to support our theoretical findings.

Federated reinforcement learning (FRL) has emerged as a promising paradigm for reducing the sample complexity of reinforcement learning tasks by exploiting information from different agents. However, when each agent interacts with a potentially different environment, little to nothing is known theoretically about the non-asymptotic performance of FRL algorithms. The lack of such results can be attributed to various technical challenges and their intricate interplay: Markovian sampling, linear function approximation, multiple local updates to save communication, heterogeneity in the reward functions and transition kernels of the agents' MDPs, and continuous state-action spaces. Moreover, in the on-policy setting, the behavior policies vary with time, further complicating the analysis. In response, we introduce FedSARSA, a novel federated on-policy reinforcement learning scheme, equipped with linear function approximation, to address these challenges and provide a comprehensive finite-time error analysis. Notably, we establish that FedSARSA converges to a policy that is near-optimal for all agents, with the extent of near-optimality proportional to the level of heterogeneity. Furthermore, we prove that FedSARSA leverages agent collaboration to enable linear speedups as the number of agents increases, which holds for both fixed and adaptive step-size configurations.

In this paper we present a novel method for efficient and effective 3D surface reconstruction in open scenes. Existing Neural Radiance Fields (NeRF) based works typically require extensive training and rendering time due to the adopted implicit representations. In contrast, 3D Gaussian splatting (3DGS) uses an explicit and discrete representation, hence the reconstructed surface is built by the huge number of Gaussian primitives, which leads to excessive memory consumption and rough surface details in sparse Gaussian areas. To address these issues, we propose Gaussian Voxel Kernel Functions (GVKF), which establish a continuous scene representation based on discrete 3DGS through kernel regression. The GVKF integrates fast 3DGS rasterization and highly effective scene implicit representations, achieving high-fidelity open scene surface reconstruction. Experiments on challenging scene datasets demonstrate the efficiency and effectiveness of our proposed GVKF, featuring with high reconstruction quality, real-time rendering speed, significant savings in storage and training memory consumption.

We develop the Itô calculus of Brownian motion, machine-checked in Lean~4 over Mathlib and the BrownianMotion package. On a bounded interval [0,T] the Itô integral is built as a Hilbert-space isometry, from a predictable-rectangle π-system through the density of simple adapted processes. Realized as a process, it is a continuous L^2 martingale. One structural identity drives this: the integral at time t is the conditional-expectation projection of its terminal value onto F_t, and from it adaptedness, the martingale property, the contraction bound, and both the terminal and time-indexed Itô isometries follow as corollaries. On this integral we prove Itô's formula for C^3 functions with bounded derivatives, including the time-dependent form df = f_x,dB + (f_t + tfrac12 f_{xx}),dt, by a discrete-to-continuous argument through weighted quadratic variation with explicit L^2 remainder bounds. We then pass from the L^2 theory to the pathwise. The integral process has an almost-surely continuous modification, and its everywhere-continuous representative is a local martingale for the null-augmented Brownian filtration; gluing the bounded-horizon representatives along the half-line yields the Itô integral as a continuous local martingale on all of R_{ge 0}, the form it takes in the classical theory. To our knowledge these are the first machine-checked constructions of the Itô integral and of Itô's formula in any proof assistant, and the first to reach a pathwise-continuous local martingale. The boundary is explicit. The L^2 integral and Itô's formula are developed on [0,T] with bounded-derivative integrands; the unrestricted C^2 formula, integrators beyond brownian motion, and right-continuity of the filtration lie outside the development.

X-ray observing facilities, such as the Chandra X-ray Observatory and the eROSITA, have detected millions of astronomical sources associated with high-energy phenomena. The arrival of photons as a function of time follows a Poisson process and can vary by orders-of-magnitude, presenting obstacles for common tasks such as source classification, physical property derivation, and anomaly detection. Previous work has either failed to directly capture the Poisson nature of the data or only focuses on Poisson rate function reconstruction. In this work, we present Poisson Process AutoDecoder (PPAD). PPAD is a neural field decoder that maps fixed-length latent features to continuous Poisson rate functions across energy band and time via unsupervised learning. PPAD reconstructs the rate function and yields a representation at the same time. We demonstrate the efficacy of PPAD via reconstruction, regression, classification and anomaly detection experiments using the Chandra Source Catalog.

This paper reformulates the Greenwood and Guner (2009) marriage and divorce model in continuous time using the HACT methods of Achdou et al. (2022). Replacing the AR(1) match quality process with an Ornstein-Uhlenbeck process yields a tridiagonal generator, reducing the computational complexity of both the value function and stationary distribution calculations from quadratic to linear in the number of grid points. The continuous-time model closely replicates the discrete-time equilibrium across all key outcomes, including the share of married households, the marriage rate, and the divorce rate, while achieving substantial gains in computation time and memory usage.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

Lion (Evolved Sign Momentum), a new optimizer discovered through program search, has shown promising results in training large AI models. It performs comparably or favorably to AdamW but with greater memory efficiency. As we can expect from the results of a random search program, Lion incorporates elements from several existing algorithms, including signed momentum, decoupled weight decay, Polak, and Nesterov momentum, but does not fit into any existing category of theoretically grounded optimizers. Thus, even though Lion appears to perform well as a general-purpose optimizer for a wide range of tasks, its theoretical basis remains uncertain. This lack of theoretical clarity limits opportunities to further enhance and expand Lion's efficacy. This work aims to demystify Lion. Based on both continuous-time and discrete-time analysis, we demonstrate that Lion is a theoretically novel and principled approach for minimizing a general loss function f(x) while enforcing a bound constraint |x|_infty leq 1/lambda. Lion achieves this through the incorporation of decoupled weight decay, where lambda represents the weight decay coefficient. Our analysis is made possible by the development of a new Lyapunov function for the Lion updates. It applies to a broader family of Lion-kappa algorithms, where the sign(cdot) operator in Lion is replaced by the subgradient of a convex function kappa, leading to the solution of a general composite optimization problem of min_x f(x) + kappa^*(x). Our findings provide valuable insights into the dynamics of Lion and pave the way for further improvements and extensions of Lion-related algorithms.

Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or in the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Pad\'e Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function. Both methods yield considerable speed-up compared with the SVD or the Newton-Schulz iteration. Experimental results on the de-correlated batch normalization and second-order vision transformer demonstrate that our methods can also achieve competitive and even slightly better performances. The code is available at https://github.com/KingJamesSong/FastDifferentiableMatSqrt{https://github.com/KingJamesSong/FastDifferentiableMatSqrt}.

Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce ``Blackout Diffusion'', which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.

In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle to efficiently capture regularities in the world, but the role of symmetry breaking is not well understood. Here, we develop a theoretical framework to study the "geometry of learning dynamics" in neural networks, and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. To build this understanding, we model the discrete learning dynamics of gradient descent using a continuous-time Lagrangian formulation, in which the learning rule corresponds to the kinetic energy and the loss function corresponds to the potential energy. Then, we identify "kinetic symmetry breaking" (KSB), the condition when the kinetic energy explicitly breaks the symmetry of the potential function. We generalize Noether's theorem known in physics to take into account KSB and derive the resulting motion of the Noether charge: "Noether's Learning Dynamics" (NLD). Finally, we apply NLD to neural networks with normalization layers and reveal how KSB introduces a mechanism of "implicit adaptive optimization", establishing an analogy between learning dynamics induced by normalization layers and RMSProp. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.

Continuous space-time video super-resolution (C-STVSR) endeavors to upscale videos simultaneously at arbitrary spatial and temporal scales, which has recently garnered increasing interest. However, prevailing methods struggle to yield satisfactory videos at out-of-distribution spatial and temporal scales. On the other hand, event streams characterized by high temporal resolution and high dynamic range, exhibit compelling promise in vision tasks. This paper presents EvEnhancer, an innovative approach that marries the unique advantages of event streams to elevate effectiveness, efficiency, and generalizability for C-STVSR. Our approach hinges on two pivotal components: 1) Event-adapted synthesis capitalizes on the spatiotemporal correlations between frames and events to discern and learn long-term motion trajectories, enabling the adaptive interpolation and fusion of informative spatiotemporal features; 2) Local implicit video transformer integrates local implicit video neural function with cross-scale spatiotemporal attention to learn continuous video representations utilized to generate plausible videos at arbitrary resolutions and frame rates. Experiments show that EvEnhancer achieves superiority on synthetic and real-world datasets and preferable generalizability on out-of-distribution scales against state-of-the-art methods. Code is available at https://github.com/W-Shuoyan/EvEnhancer.

The analysis of non-stationary time-series data requires insight into its local and global patterns with physical interpretability. However, traditional smoothing algorithms, such as B-splines, Savitzky-Golay filtering, and Empirical Mode Decomposition (EMD), lack the ability to perform parametric optimization with guaranteed continuity. In this paper, we propose Functional Continuous Decomposition (FCD), a JAX-accelerated framework that performs parametric, continuous optimization on a wide range of mathematical functions. By using Levenberg-Marquardt optimization to achieve up to C^1 continuous fitting, FCD transforms raw time-series data into M modes that capture different temporal patterns from short-term to long-term trends. Applications of FCD include physics, medicine, financial analysis, and machine learning, where it is commonly used for the analysis of signal temporal patterns, optimized parameters, derivatives, and integrals of decomposition. Furthermore, FCD can be applied for physical analysis and feature extraction with an average SRMSE of 0.735 per segment and a speed of 0.47s on full decomposition of 1,000 points. Finally, we demonstrate that a Convolutional Neural Network (CNN) enhanced with FCD features, such as optimized function values, parameters, and derivatives, achieved 16.8% faster convergence and 2.5% higher accuracy over a standard CNN.

This paper explores the expressive power of deep neural networks through the framework of function compositions. We demonstrate that the repeated compositions of a single fixed-size ReLU network exhibit surprising expressive power, despite the limited expressive capabilities of the individual network itself. Specifically, we prove by construction that L_2circ g^{circ r}circ mathcal{L}_1 can approximate 1-Lipschitz continuous functions on [0,1]^d with an error O(r^{-1/d}), where g is realized by a fixed-size ReLU network, mathcal{L}_1 and L_2 are two affine linear maps matching the dimensions, and g^{circ r} denotes the r-times composition of g. Furthermore, we extend such a result to generic continuous functions on [0,1]^d with the approximation error characterized by the modulus of continuity. Our results reveal that a continuous-depth network generated via a dynamical system has immense approximation power even if its dynamics function is time-independent and realized by a fixed-size ReLU network.

Effective spatio-temporal representation is fundamental to modeling, understanding, and predicting dynamics in videos. The atomic unit of a video, the pixel, traces a continuous 3D trajectory over time, serving as the primitive element of dynamics. Based on this principle, we propose representing any video as a Trajectory Field: a dense mapping that assigns a continuous 3D trajectory function of time to each pixel in every frame. With this representation, we introduce Trace Anything, a neural network that predicts the entire trajectory field in a single feed-forward pass. Specifically, for each pixel in each frame, our model predicts a set of control points that parameterizes a trajectory (i.e., a B-spline), yielding its 3D position at arbitrary query time instants. We trained the Trace Anything model on large-scale 4D data, including data from our new platform, and our experiments demonstrate that: (i) Trace Anything achieves state-of-the-art performance on our new benchmark for trajectory field estimation and performs competitively on established point-tracking benchmarks; (ii) it offers significant efficiency gains thanks to its one-pass paradigm, without requiring iterative optimization or auxiliary estimators; and (iii) it exhibits emergent abilities, including goal-conditioned manipulation, motion forecasting, and spatio-temporal fusion. Project page: https://trace-anything.github.io/.

Aligning large language models with human feedback at inference time has received increasing attention due to its flexibility. Existing methods rely on generating multiple responses from the base policy for search using a reward model, which can be considered as searching in a discrete response space. However, these methods struggle to explore informative candidates when the base policy is weak or the candidate set is small, resulting in limited effectiveness. In this paper, to address this problem, we propose Simple Energy Adaptation (SEA), a simple yet effective algorithm for inference-time alignment. In contrast to expensive search over the discrete space, SEA directly adapts original responses from the base policy toward the optimal one via gradient-based sampling in continuous latent space. Specifically, SEA formulates inference as an iterative optimization procedure on an energy function over actions in the continuous space defined by the optimal policy, enabling simple and effective alignment. For instance, despite its simplicity, SEA outperforms the second-best baseline with a relative improvement of up to 77.51% on AdvBench and 16.36% on MATH. Our code is publicly available at https://github.com/yuanyige/sea

The emergence of Multimodal Large Language Models (MLLMs) has driven significant advances in Graphical User Interface (GUI) agent capabilities. Nevertheless, existing GUI agent training and inference techniques still suffer from a dilemma for reasoning designs, ineffective reward, and visual noise. To address these issues, we introduce UI-AGILE, a comprehensive framework enhancing GUI agents at both the training and inference stages. For training, we propose a suite of improvements to the Supervised Fine-Tuning (SFT) process: 1) a Continuous Reward function to incentivize high-precision grounding; 2) a "Simple Thinking" reward to balance planning with speed and grounding accuracy; and 3) a Cropping-based Resampling strategy to mitigate the sparse reward problem and improve learning on complex tasks. For inference, we present Decomposed Grounding with Selection, a novel method that dramatically improves grounding accuracy on high-resolution displays by breaking the image into smaller, manageable parts. Experiments show that UI-AGILE achieves the state-of-the-art performance on two benchmarks ScreenSpot-Pro and ScreenSpot-v2. For instance, using both our proposed training and inference enhancement methods brings 23% grounding accuracy improvement over the best baseline on ScreenSpot-Pro.

This paper addresses the task of space-time video super-resolution (ST-VSR). Existing methods generally suffer from inaccurate motion estimation and motion compensation (MEMC) problems for large motions. Inspired by recent progress in physics-informed neural networks, we model the challenges of MEMC in ST-VSR as a mapping between two continuous function spaces. Specifically, our approach transforms independent low-resolution representations in the coarse-grained continuous function space into refined representations with enriched spatiotemporal details in the fine-grained continuous function space. To achieve efficient and accurate MEMC, we design a Galerkin-type attention function to perform frame alignment and temporal interpolation. Due to the linear complexity of the Galerkin-type attention mechanism, our model avoids patch partitioning and offers global receptive fields, enabling precise estimation of large motions. The experimental results show that the proposed method surpasses state-of-the-art techniques in both fixed-size and continuous space-time video super-resolution tasks.

Irregularly-sampled time series occur in many domains including healthcare. They can be challenging to model because they do not naturally yield a fixed-dimensional representation as required by many standard machine learning models. In this paper, we consider irregular sampling from the perspective of missing data. We model observed irregularly-sampled time series data as a sequence of index-value pairs sampled from a continuous but unobserved function. We introduce an encoder-decoder framework for learning from such generic indexed sequences. We propose learning methods for this framework based on variational autoencoders and generative adversarial networks. For continuous irregularly-sampled time series, we introduce continuous convolutional layers that can efficiently interface with existing neural network architectures. Experiments show that our models are able to achieve competitive or better classification results on irregularly-sampled multivariate time series compared to recent RNN models while offering significantly faster training times.

Large language models (LLMs) have shown remarkable in-context learning (ICL) capabilities on textual data. We explore whether these capabilities can be extended to continuous vectors from diverse domains, obtained from black-box pretrained encoders. By aligning input data with an LLM's embedding space through lightweight projectors, we observe that LLMs can effectively process and learn from these projected vectors, which we term Vector-ICL. In particular, we find that pretraining projectors with general language modeling objectives enables Vector-ICL, while task-specific finetuning further enhances performance. In our experiments across various tasks and modalities, including text reconstruction, numerical function regression, text classification, summarization, molecule captioning, time-series classification, graph classification, and fMRI decoding, Vector-ICL often surpasses both few-shot ICL and domain-specific model or tuning. We further conduct analyses and case studies, indicating the potential of LLMs to process vector representations beyond traditional token-based paradigms.

Radiance of real-world scenes typically spans a much wider dynamic range than what standard cameras can capture. While conventional HDR methods merge alternating-exposure frames, these approaches are inherently constrained to 2D pixel-level alignment, often leading to ghosting artifacts and temporal inconsistency in dynamic scenes. To address these limitations, we present HDR-NSFF, a paradigm shift from 2D-based merging to 4D spatio-temporal modeling. Our framework reconstructs dynamic HDR radiance fields from alternating-exposure monocular videos by representing the scene as a continuous function of space and time, and is compatible with both neural radiance field and 4D Gaussian Splatting (4DGS) based dynamic representations. This unified end-to-end pipeline explicitly models HDR radiance, 3D scene flow, geometry, and tone-mapping, ensuring physical plausibility and global coherence. We further enhance robustness by (i) extending semantic-based optical flow with DINO features to achieve exposure-invariant motion estimation, and (ii) incorporating a generative prior as a regularizer to compensate for limited observation in monocular captures and saturation-induced information loss. To evaluate HDR space-time view synthesis, we present the first real-world HDR-GoPro dataset specifically designed for dynamic HDR scenes. Experiments demonstrate that HDR-NSFF recovers fine radiance details and coherent dynamics even under challenging exposure variations, thereby achieving state-of-the-art performance in novel space-time view synthesis. Project page: https://shin-dong-yeon.github.io/HDR-NSFF/

This script offers an implementation-oriented introduction to deep learning methods for solving and estimating high-dimensional dynamic stochastic models in economics and finance. Its starting point is the curse of dimensionality: heterogeneous-agent economies, overlapping-generations models with aggregate risk, continuous-time models with occasionally binding constraints, climate-economy models, and macro-finance environments with many assets and frictions generate state and parameter spaces that strain classical tensor-product grid methods. The exposition is organized around four complementary methodologies. Deep Equilibrium Nets embed discrete-time equilibrium conditions into neural-network loss functions. Physics-Informed Neural Networks approximate continuous-time Hamilton--Jacobi--Bellman, Kolmogorov forward, and related partial differential equations. Deep surrogate models provide fast, differentiable approximations to expensive structural models, while Gaussian processes add a probabilistic layer that quantifies approximation uncertainty; together they support estimation, sensitivity analysis, and constrained policy design. Gaussian-process-based dynamic programming, combined with active learning and dimension reduction, extends value-function iteration to very large continuous state spaces. Applications span representative-agent and international real business cycle models, overlapping-generations and heterogeneous-agent economies, continuous-time macro-finance, structural estimation by simulated method of moments, and climate economics under uncertainty. Companion notebooks in TensorFlow and PyTorch invite hands-on experimentation. These notes are a deliberately subjective and inevitably incomplete snapshot of a rapidly evolving field, aimed at equipping PhD students and researchers to engage with this frontier hands-on.

Continuous-time neural processes are performant sequential decision-makers that are built by differential equations (DE). However, their expressive power when they are deployed on computers is bottlenecked by numerical DE solvers. This limitation has significantly slowed down the scaling and understanding of numerous natural physical phenomena such as the dynamics of nervous systems. Ideally, we would circumvent this bottleneck by solving the given dynamical system in closed form. This is known to be intractable in general. Here, we show it is possible to closely approximate the interaction between neurons and synapses -- the building blocks of natural and artificial neural networks -- constructed by liquid time-constant networks (LTCs) efficiently in closed-form. To this end, we compute a tightly-bounded approximation of the solution of an integral appearing in LTCs' dynamics, that has had no known closed-form solution so far. This closed-form solution substantially impacts the design of continuous-time and continuous-depth neural models; for instance, since time appears explicitly in closed-form, the formulation relaxes the need for complex numerical solvers. Consequently, we obtain models that are between one and five orders of magnitude faster in training and inference compared to differential equation-based counterparts. More importantly, in contrast to ODE-based continuous networks, closed-form networks can scale remarkably well compared to other deep learning instances. Lastly, as these models are derived from liquid networks, they show remarkable performance in time series modeling, compared to advanced recurrent models.

We propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and enable flexible, high-fidelity models of discrete events that are localized in continuous time and space. Central to our approach is a combination of continuous-time neural networks with two novel neural architectures, i.e., Jump and Attentive Continuous-time Normalizing Flows. This approach allows us to learn complex distributions for both the spatial and temporal domain and to condition non-trivially on the observed event history. We validate our models on data sets from a wide variety of contexts such as seismology, epidemiology, urban mobility, and neuroscience.

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.

Continuous normalizing flows (CNFs) construct invertible mappings between an arbitrary complex distribution and an isotropic Gaussian distribution using Neural Ordinary Differential Equations (neural ODEs). It has not been tractable on large datasets due to the incremental complexity of the neural ODE training. Optimal Transport theory has been applied to regularize the dynamics of the ODE to speed up training in recent works. In this paper, a temporal optimization is proposed by optimizing the evolutionary time for forward propagation of the neural ODE training. In this appoach, we optimize the network weights of the CNF alternately with evolutionary time by coordinate descent. Further with temporal regularization, stability of the evolution is ensured. This approach can be used in conjunction with the original regularization approach. We have experimentally demonstrated that the proposed approach can significantly accelerate training without sacrifying performance over baseline models.

We introduce a new class of time-continuous recurrent neural network models. Instead of declaring a learning system's dynamics by implicit nonlinearities, we construct networks of linear first-order dynamical systems modulated via nonlinear interlinked gates. The resulting models represent dynamical systems with varying (i.e., liquid) time-constants coupled to their hidden state, with outputs being computed by numerical differential equation solvers. These neural networks exhibit stable and bounded behavior, yield superior expressivity within the family of neural ordinary differential equations, and give rise to improved performance on time-series prediction tasks. To demonstrate these properties, we first take a theoretical approach to find bounds over their dynamics and compute their expressive power by the trajectory length measure in latent trajectory space. We then conduct a series of time-series prediction experiments to manifest the approximation capability of Liquid Time-Constant Networks (LTCs) compared to classical and modern RNNs. Code and data are available at https://github.com/raminmh/liquid_time_constant_networks

Sequential VAEs have been successfully considered for many high-dimensional time series modelling problems, with many variant models relying on discrete-time mechanisms such as recurrent neural networks (RNNs). On the other hand, continuous-time methods have recently gained attraction, especially in the context of irregularly-sampled time series, where they can better handle the data than discrete-time methods. One such class are Gaussian process variational autoencoders (GPVAEs), where the VAE prior is set as a Gaussian process (GP). However, a major limitation of GPVAEs is that it inherits the cubic computational cost as GPs, making it unattractive to practioners. In this work, we leverage the equivalent discrete state space representation of Markovian GPs to enable linear time GPVAE training via Kalman filtering and smoothing. We show on a variety of high-dimensional temporal and spatiotemporal tasks that our method performs favourably compared to existing approaches whilst being computationally highly scalable.

In this paper we show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function F:I^2to I is continuous. As a consequence, we obtain a finer characterization of quasi-arithmetic means than the classical results of Aczél, Kolmogoroff, Nagumo and de Finetti.

Discovering the underlying relationships among variables from temporal observations has been a longstanding challenge in numerous scientific disciplines, including biology, finance, and climate science. The dynamics of such systems are often best described using continuous-time stochastic processes. Unfortunately, most existing structure learning approaches assume that the underlying process evolves in discrete-time and/or observations occur at regular time intervals. These mismatched assumptions can often lead to incorrect learned structures and models. In this work, we introduce a novel structure learning method, SCOTCH, which combines neural stochastic differential equations (SDE) with variational inference to infer a posterior distribution over possible structures. This continuous-time approach can naturally handle both learning from and predicting observations at arbitrary time points. Theoretically, we establish sufficient conditions for an SDE and SCOTCH to be structurally identifiable, and prove its consistency under infinite data limits. Empirically, we demonstrate that our approach leads to improved structure learning performance on both synthetic and real-world datasets compared to relevant baselines under regular and irregular sampling intervals.

A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

We investigate the first-order differential calculus over extended metric-topological measure spaces. The latter are quartets mathbb X=(X,τ,{sf d},mathfrak m), given by an extended metric space (X,{sf d}) together with a weaker topology τ (satisfying suitable compatibility conditions) and a finite Radon measure mathfrak m on (X,τ). The class of extended metric-topological measure spaces encompasses all metric measure spaces and many infinite-dimensional metric-measure structures, such as abstract Wiener spaces. In this framework, we study the following classes of objects: - The Banach algebra {rm Lip}_b(X,τ,{sf d}) of bounded τ-continuous {sf d}-Lipschitz functions on X. - Several notions of Lipschitz derivations on X, defined in duality with {rm Lip}_b(X,τ,{sf d}). - The metric Sobolev space W^{1,p}(mathbb X), defined in duality with Lipschitz derivations on X. Inter alia, we generalise both Weaver's and Di Marino's theories of Lipschitz derivations to the extended setting, and we discuss their connections. We also introduce a Sobolev space W^{1,p}(mathbb X) via an integration-by-parts formula, along the lines of Di Marino's notion of Sobolev space, and we prove its equivalence with other approaches, studied in the extended setting by Ambrosio, Erbar and Savaré. En route, we obtain some results of independent interest, among which are: - A Lipschitz-constant-preserving extension result for τ-continuous {sf d}-Lipschitz functions. - A novel and rather robust strategy for proving the equivalence of Sobolev-type spaces defined via an integration-by-parts formula and those obtained with a relaxation procedure. - A new description of an isometric predual of the metric Sobolev space W^{1,p}(mathbb X).

Generating continuous-time, continuous-space stochastic processes (e.g., videos, weather forecasts) conditioned on partial observations (e.g., first and last frames) is a fundamental challenge. Existing approaches, (e.g., diffusion models), suffer from key limitations: (1) noise-to-data evolution fails to capture structural similarity between states close in physical time and has unstable integration in low-step regimes; (2) random noise injected is insensitive to the physical process's time elapsed, resulting in incorrect dynamics; (3) they overlook conditioning on arbitrary subsets of states (e.g., irregularly sampled timesteps, future observations). We propose ABC: Any-Subset Autoregressive Models via Non-Markovian Diffusion Bridges in Continuous Time and Space. Crucially, we model the process with one continual SDE whose time variable and intermediate states track the real time and process states. This has provable advantages: (1) the starting point for generating future states is the already-close previous state, rather than uninformative noise; (2) random noise injection scales with physical time elapsed, encouraging physically plausible dynamics with similar time-adjacent states. We derive SDE dynamics via changes-of-measure on path space, yielding another advantage: (3) path-dependent conditioning on arbitrary subsets of the state history and/or future. To learn these dynamics, we derive a path- and time-dependent extension of denoising score matching. Our experiments show ABC's superiority to competing methods on multiple domains, including video generation and weather forecasting.

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.

Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better generalization to the dynamical system. The architecture is empirically evaluated against discrete models, specifically Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs), and other continuous-depth variants, including Augmented Neural ODE (ANODE), across four distinct dynamical systems: the Lotka-Volterra model, the forced Duffing oscillator, the Van der Pol oscillator, and the Lorenz system. To rigorously assess generalization and robustness, curriculum and ensemble learning are used to evaluate the model's convergence by estimating confidence intervals across different ensemble models. The empirical results demonstrate that the FNODE architecture achieves better generalization while exhibiting remarkable convergence stability.

Commuting families of contractions or contractive C_{0}-semigroups on Hilbert spaces often fail to admit power dilations resp, simultaneous unitary dilations which are themselves commutative (see [45, 13, 15]). In the non-commutative setting, Sz.-Nagy [60] and Bożejko [5] provided means to dilate arbitrary families of contractions. The present work extends these discrete-time results to families {T_{i}}_{i in I} of contractive C_{0}-semigroups. We refer to these dilations as continuous-time free unitary dilations and present three distinct approaches to obtain them: 1) An explicit derivation applicable to semigroups that arise as interpolations; 2) A full proof with an explicit construction, via the theory of co-generators à la Słociński [54, 55]; and 3) A second full proof based on the abstract structure of semigroups, which admits a natural reformulation to semigroups defined over topological free products of R_{geq 0} and leads to various residuality results. In 2) a IInd free dilation theorem for topologised index sets is developed via a reformulation of the Trotter--Kato theorem for co-generators. As an application of this we demonstrate how evolution families can be reduced to continuously monitored processes subject to temporal change, à la the quantum Zeno effect [22, 23, 24, 30, 37].

Conventional neural architectures for sequential data present important limitations. Recurrent networks suffer from exploding and vanishing gradients, small effective memory horizons, and must be trained sequentially. Convolutional networks are unable to handle sequences of unknown size and their memory horizon must be defined a priori. In this work, we show that all these problems can be solved by formulating convolutional kernels in CNNs as continuous functions. The resulting Continuous Kernel Convolution (CKConv) allows us to model arbitrarily long sequences in a parallel manner, within a single operation, and without relying on any form of recurrence. We show that Continuous Kernel Convolutional Networks (CKCNNs) obtain state-of-the-art results in multiple datasets, e.g., permuted MNIST, and, thanks to their continuous nature, are able to handle non-uniformly sampled datasets and irregularly-sampled data natively. CKCNNs match or perform better than neural ODEs designed for these purposes in a faster and simpler manner.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

Extending discrete-time causal Prior-data Fitted Networks for time series to continuous time invites writing the mechanism as a stochastic differential equation (SDE) -- but if the SDE is integrated once per observation gap, the trajectory law depends on when it is observed, and the prior remains a discrete-time Markov model in SDE clothing. We propose a precise continuity criterion -- trajectory-law invariance to the observation schedule -- together with a three-tier taxonomy (discrete; naive observation-grid integration; fine-grid integration with decoupled observation) and a construction realising the top tier on a random DAG with OU or small-MLP nonlinear drifts, irregular observation schedules, and hard / soft / time-varying interventions. A 2 times 2 encoder times integrator ablation, run independently on a linear and a nonlinear prior, finds fine-grid integration beats naive on 8/8 cells (sign-consistency p < 1/256) with the gap growing as the eval grid refines; the encoder axis is null with fine integration but time-aware-leading with naive. We release the prior and a preliminary zero-shot protocol on pharmacokinetic and physical-system data.

This paper presents a continuous-time optimal control framework for the generation of reference trajectories in driving scenarios with uncertainty. A previous work presented a discrete-time stochastic generator for autonomous vehicles; those results are extended to continuous time to ensure the robustness of the generator in a real-time setting. We show that the stochastic model in continuous time can capture the uncertainty of information by producing better results, limiting the risk of violating the problem's constraints compared to a discrete approach. Dynamic solvers provide faster computation and the continuous-time model is more robust to a wider variety of driving scenarios than the discrete-time model, as it can handle further time horizons, which allows trajectory planning outside the framework of urban driving scenarios.

Learning the continuous dynamics of a system from snapshots of its temporal marginals is a problem which appears throughout natural sciences and machine learning, including in quantum systems, single-cell biological data, and generative modeling. In these settings, we assume access to cross-sectional samples that are uncorrelated over time, rather than full trajectories of samples. In order to better understand the systems under observation, we would like to learn a model of the underlying process that allows us to propagate samples in time and thereby simulate entire individual trajectories. In this work, we propose Action Matching, a method for learning a rich family of dynamics using only independent samples from its time evolution. We derive a tractable training objective, which does not rely on explicit assumptions about the underlying dynamics and does not require back-propagation through differential equations or optimal transport solvers. Inspired by connections with optimal transport, we derive extensions of Action Matching to learn stochastic differential equations and dynamics involving creation and destruction of probability mass. Finally, we showcase applications of Action Matching by achieving competitive performance in a diverse set of experiments from biology, physics, and generative modeling.

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this two part treatise, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct classes of algorithms, namely continuous time and discrete time models. The resulting neural networks form a new class of data-efficient universal function approximators that naturally encode any underlying physical laws as prior information. In this first part, we demonstrate how these networks can be used to infer solutions to partial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters.

We consider the question of learning Q-function in a sample efficient manner for reinforcement learning with continuous state and action spaces under a generative model. If Q-function is Lipschitz continuous, then the minimal sample complexity for estimating ε-optimal Q-function is known to scale as Ω(1{ε^{d_1+d_2 +2}}) per classical non-parametric learning theory, where d_1 and d_2 denote the dimensions of the state and action spaces respectively. The Q-function, when viewed as a kernel, induces a Hilbert-Schmidt operator and hence possesses square-summable spectrum. This motivates us to consider a parametric class of Q-functions parameterized by its "rank" r, which contains all Lipschitz Q-functions as r to infty. As our key contribution, we develop a simple, iterative learning algorithm that finds ε-optimal Q-function with sample complexity of O(1{ε^{max(d_1, d_2)+2}}) when the optimal Q-function has low rank r and the discounting factor γ is below a certain threshold. Thus, this provides an exponential improvement in sample complexity. To enable our result, we develop a novel Matrix Estimation algorithm that faithfully estimates an unknown low-rank matrix in the ell_infty sense even in the presence of arbitrary bounded noise, which might be of interest in its own right. Empirical results on several stochastic control tasks confirm the efficacy of our "low-rank" algorithms.

Recent work on the sequence universality of State Space Models (SSMs) has introduced efficient, maximally expressive continuous-time approaches for time-series modelling. While these works focus on discriminative settings, we extend this perspective to generative time-series modelling by proving that maximally expressive Structured Linear Controlled Differential Equations (SLiCEs) are universal time-series generators, in the sense that they can approximate the induced path laws of continuous causal pushforwards on compact latent sets in W_infty. Building on these theoretical results, we propose Generative SLiCEs (G-SLiCEs), a maximally expressive continuous-time model for flow matching on path-space. Empirically, we show that expressivity improves performance in probabilistic forecasting and downstream tasks, while retaining the advantages of continuous-time models such as generalising to arbitrary observation grids. This is particularly beneficial for irregular grids, where fixed-grid models often struggle.

We describe a library of mathematical finance built in the Lean 4 proof assistant, on top of Mathlib and the BrownianMotion package. It is broad: more than two hundred sorry-free theorems across eleven areas, from the measure-theoretic foundations of continuous-time stochastic calculus through derivative pricing to applied risk, portfolio, and fixed-income theory, and, to our knowledge, the most comprehensive machine-checked development of mathematical finance to date. Breadth is the setting, not the point. Two things make it more than a catalogue. It reaches into the continuous theory far enough to construct the L2 Itô integral as a bounded linear isometry and to derive, rather than assume, the risk-neutral pricing measure. And it audits its own faithfulness: every result is classified by how its Lean statement relates to the mathematics it claims, and a build-enforced gate pins the axioms each proof actually uses, so a reader can see precisely what has been proved and what has only been proved under added hypotheses. We close with a candid finding: a formal base over classical financial mathematics yields certified unification of known results rather than new financial theory. The contribution is therefore methodological and infrastructural, reusable verified foundations for mathematical finance, together with the faithfulness audit.

We propose a reinforcement learning (RL) framework under a broad class of risk objectives, characterized by convex scoring functions. This class covers many common risk measures, such as variance, Expected Shortfall, entropic Value-at-Risk, and mean-risk utility. To resolve the time-inconsistency issue, we consider an augmented state space and an auxiliary variable and recast the problem as a two-state optimization problem. We propose a customized Actor-Critic algorithm and establish some theoretical approximation guarantees. A key theoretical contribution is that our results do not require the Markov decision process to be continuous. Additionally, we propose an auxiliary variable sampling method inspired by the alternating minimization algorithm, which is convergent under certain conditions. We validate our approach in simulation experiments with a financial application in statistical arbitrage trading, demonstrating the effectiveness of the algorithm.

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant Agda.

Wider adoption of neural networks in many critical domains such as finance and healthcare is being hindered by the need to explain their predictions and to impose additional constraints on them. Monotonicity constraint is one of the most requested properties in real-world scenarios and is the focus of this paper. One of the oldest ways to construct a monotonic fully connected neural network is to constrain signs on its weights. Unfortunately, this construction does not work with popular non-saturated activation functions as it can only approximate convex functions. We show this shortcoming can be fixed by constructing two additional activation functions from a typical unsaturated monotonic activation function and employing each of them on the part of neurons. Our experiments show this approach of building monotonic neural networks has better accuracy when compared to other state-of-the-art methods, while being the simplest one in the sense of having the least number of parameters, and not requiring any modifications to the learning procedure or post-learning steps. Finally, we prove it can approximate any continuous monotone function on a compact subset of R^n.

Learning accurate predictive models of real-world dynamic phenomena (e.g., climate, biological) remains a challenging task. One key issue is that the data generated by both natural and artificial processes often comprise time series that are irregularly sampled and/or contain missing observations. In this work, we propose the Neural Continuous-Discrete State Space Model (NCDSSM) for continuous-time modeling of time series through discrete-time observations. NCDSSM employs auxiliary variables to disentangle recognition from dynamics, thus requiring amortized inference only for the auxiliary variables. Leveraging techniques from continuous-discrete filtering theory, we demonstrate how to perform accurate Bayesian inference for the dynamic states. We propose three flexible parameterizations of the latent dynamics and an efficient training objective that marginalizes the dynamic states during inference. Empirical results on multiple benchmark datasets across various domains show improved imputation and forecasting performance of NCDSSM over existing models.

Continuous convolution has recently gained prominence due to its ability to handle irregularly sampled data and model long-term dependency. Also, the promising experimental results of using large convolutional kernels have catalyzed the development of continuous convolution since they can construct large kernels very efficiently. Leveraging neural networks, more specifically multilayer perceptrons (MLPs), is by far the most prevalent approach to implementing continuous convolution. However, there are a few drawbacks, such as high computational costs, complex hyperparameter tuning, and limited descriptive power of filters. This paper suggests an alternative approach to building a continuous convolution without neural networks, resulting in more computationally efficient and improved performance. We present self-moving point representations where weight parameters freely move, and interpolation schemes are used to implement continuous functions. When applied to construct convolutional kernels, the experimental results have shown improved performance with drop-in replacement in the existing frameworks. Due to its lightweight structure, we are first to demonstrate the effectiveness of continuous convolution in a large-scale setting, e.g., ImageNet, presenting the improvements over the prior arts. Our code is available on https://github.com/sangnekim/SMPConv

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors x_i, i=1,dots,m (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.

Time series prototype learning is fundamentally challenged by observational ambiguity. Discrete architectures fail to resolve this, as they lack the capacity to decouple stochastic noise from continuous dynamics. Furthermore, rigid closed-set assumptions fail to capture unseen diversity. To address these limitations, we propose a hierarchical ordinary differential equation clustering network, which utilizes neural ordinary differential equation to model latent state evolution as a continuous integral curve. This formulation enforces temporal continuity to effectively disentangle smooth feature trends from stochastic noise, while our adaptive hierarchical mechanism autonomously determines the appropriate number of prototypes without rigid prior constraints. Validated on the early link failure detection task with irregularly sampled time series, the proposed method effectively extracts underlying physical prototypes, thereby enabling robust failure detection. Our code is available at https://github.com/NJ-LNN/Hierarchical-ODE.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood

Nonlinear ordinary differential equations (ODEs) are powerful tools for modeling real-world dynamical systems. However, propagating initial state uncertainty through nonlinear dynamics, especially when the ODE is unknown and learned from data, remains a major challenge. This paper introduces a novel continuum dynamics perspective for model learning that enables formal uncertainty propagation by constructing Taylor series approximations of probabilistic events. We establish sufficient conditions for the soundness of the approach and prove its asymptotic convergence. Empirical results demonstrate the framework's effectiveness, particularly when predicting rare events.

A recurrent neural network (RNN) is a widely used deep-learning network for dealing with sequential data. Imitating a dynamical system, an infinite-width RNN can approximate any open dynamical system in a compact domain. In general, deep networks with bounded widths are more effective than wide networks in practice; however, the universal approximation theorem for deep narrow structures has yet to be extensively studied. In this study, we prove the universality of deep narrow RNNs and show that the upper bound of the minimum width for universality can be independent of the length of the data. Specifically, we show that a deep RNN with ReLU activation can approximate any continuous function or L^p function with the widths d_x+d_y+2 and max{d_x+1,d_y}, respectively, where the target function maps a finite sequence of vectors in R^{d_x} to a finite sequence of vectors in R^{d_y}. We also compute the additional width required if the activation function is tanh or more. In addition, we prove the universality of other recurrent networks, such as bidirectional RNNs. Bridging a multi-layer perceptron and an RNN, our theory and proof technique can be an initial step toward further research on deep RNNs.

Recently, there have been numerous attempts to enhance the sample efficiency of off-policy reinforcement learning (RL) agents when interacting with the environment, including architecture improvements and new algorithms. Despite these advances, they overlook the potential of directly constraining the initial representations of the input data, which can intuitively alleviate the distribution shift issue and stabilize training. In this paper, we introduce the Tanh function into the initial layer to fulfill such a constraint. We theoretically unpack the convergence property of the temporal difference learning with the Tanh function under linear function approximation. Motivated by theoretical insights, we present our Constrained Initial Representations framework, tagged CIR, which is made up of three components: (i) the Tanh activation along with normalization methods to stabilize representations; (ii) the skip connection module to provide a linear pathway from the shallow layer to the deep layer; (iii) the convex Q-learning that allows a more flexible value estimate and mitigates potential conservatism. Empirical results show that CIR exhibits strong performance on numerous continuous control tasks, even being competitive or surpassing existing strong baseline methods.

Temporal point processes are the dominant paradigm for modeling sequences of events happening at irregular intervals. The standard way of learning in such models is by estimating the conditional intensity function. However, parameterizing the intensity function usually incurs several trade-offs. We show how to overcome the limitations of intensity-based approaches by directly modeling the conditional distribution of inter-event times. We draw on the literature on normalizing flows to design models that are flexible and efficient. We additionally propose a simple mixture model that matches the flexibility of flow-based models, but also permits sampling and computing moments in closed form. The proposed models achieve state-of-the-art performance in standard prediction tasks and are suitable for novel applications, such as learning sequence embeddings and imputing missing data.

This paper provides an introduction and overview of recent work on control barrier functions and their use to verify and enforce safety properties in the context of (optimization based) safety-critical controllers. We survey the main technical results and discuss applications to several domains including robotic systems.

We present Manifold Diffusion Fields (MDF), an approach to learn generative models of continuous functions defined over Riemannian manifolds. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. Empirical results on several datasets and manifolds show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.

Probabilistic completeness is an important property in motion planning. Although it has been established with clear assumptions for geometric planners, the panorama of completeness results for kinodynamic planners is still incomplete, as most existing proofs rely on strong assumptions that are difficult, if not impossible, to verify on practical systems. In this paper, we focus on an important class of kinodynamic planners, namely those that interpolate trajectories in the state space. We provide a proof of probabilistic completeness for these planners under assumptions that can be readily verified from the system's equations of motion and the user-defined interpolation function. Our proof relies crucially on a property of interpolated trajectories, termed second-order continuity (SOC), which we show is tightly related to the ability of a planner to benefit from denser sampling. We analyze the impact of this property in simulations on a low-torque pendulum. Our results show that a simple RRT using a second-order continuous interpolation swiftly finds solution, while it is impossible for the same planner using standard Bezier curves (which are not SOC) to find any solution.

Continuous-time event sequences play a vital role in real-world domains such as healthcare, finance, online shopping, social networks, and so on. To model such data, temporal point processes (TPPs) have emerged as the most natural and competitive models, making a significant impact in both academic and application communities. Despite the emergence of many powerful models in recent years, there hasn't been a central benchmark for these models and future research endeavors. This lack of standardization impedes researchers and practitioners from comparing methods and reproducing results, potentially slowing down progress in this field. In this paper, we present EasyTPP, the first central repository of research assets (e.g., data, models, evaluation programs, documentations) in the area of event sequence modeling. Our EasyTPP makes several unique contributions to this area: a unified interface of using existing datasets and adding new datasets; a wide range of evaluation programs that are easy to use and extend as well as facilitate reproducible research; implementations of popular neural TPPs, together with a rich library of modules by composing which one could quickly build complex models. All the data and implementation can be found at https://github.com/ant-research/EasyTemporalPointProcess. We will actively maintain this benchmark and welcome contributions from other researchers and practitioners. Our benchmark will help promote reproducible research in this field, thus accelerating research progress as well as making more significant real-world impacts.

kooplearn is a machine-learning library that implements linear, kernel, and deep-learning estimators of dynamical operators and their spectral decompositions. kooplearn can model both discrete-time evolution operators (Koopman/Transfer) and continuous-time infinitesimal generators. By learning these operators, users can analyze dynamical systems via spectral methods, derive data-driven reduced-order models, and forecast future states and observables. kooplearn's interface is compliant with the scikit-learn API, facilitating its integration into existing machine learning and data science workflows. Additionally, kooplearn includes curated benchmark datasets to support experimentation, reproducibility, and the fair comparison of learning algorithms. The software is available at https://github.com/Machine-Learning-Dynamical-Systems/kooplearn.

We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation of continuous-time models (in the form of SDEs) for SAM and two of its variants, both for the full-batch and mini-batch settings. We demonstrate that these SDEs are rigorous approximations of the real discrete-time algorithms (in a weak sense, scaling linearly with the learning rate). Using these models, we then offer an explanation of why SAM prefers flat minima over sharp ones~--~by showing that it minimizes an implicitly regularized loss with a Hessian-dependent noise structure. Finally, we prove that SAM is attracted to saddle points under some realistic conditions. Our theoretical results are supported by detailed experiments.

We introduce a new functional space U designed to contain all classical arithmetic functions (Mobius, von Mangoldt, Euler phi, divisor functions, Dirichlet characters, etc.). The norm of U combines a Hilbert-type component, based on square summability of Dirichlet coefficients for every s > 1, with a Banach component controlling logarithmic averages of partial sums. We prove that U is a complete Banach space which embeds continuously all standard Hilbert spaces of Dirichlet series and allows natural actions of Dirichlet convolution and shift operators. This framework provides a unified analytic setting for classical and modern problems in multiplicative number theory.

We investigate causal computations taking sequences of inputs to sequences of outputs where the nth output depends on the first n inputs only. We model these in category theory via a construction taking a Cartesian category C to another category St(C) with a novel trace-like operation called "delayed trace", which misses yanking and dinaturality axioms of the usual trace. The delayed trace operation provides a feedback mechanism in St(C) with an implicit guardedness guarantee. When C is equipped with a Cartesian differential operator, we construct a differential operator for St(C) using an abstract version of backpropagation through time, a technique from machine learning based on unrolling of functions. This obtains a swath of properties for backpropagation through time, including a chain rule and Schwartz theorem. Our differential operator is also able to compute the derivative of a stateful network without requiring the network to be unrolled.

We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families {S_{i}}_{i in I} of contractions on a Hilbert space H, to commuting families {T_{i}}_{i in I} of contractive C_{0}-semigroups on L^{2}(prod_{i in I}T) otimes H. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott's construction (1970), we then demonstrate for d in N with d geq 3 the existence of commuting families {T_{i}}_{i=1}^{d} of contractive C_{0}-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt. the topology of uniform wot-convergence on compact subsets of R_{geq 0}^{d} of non-unitarily dilatable and non-unitarily approximable d-parameter contractive C_{0}-semigroups on separable infinite-dimensional Hilbert spaces for each d geq 3. Similar results are also developed for d-tuples of commuting contractions. And by building on the counter-examples of Varopoulos--Kaijser (1973--74), a 0--1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, viz. that `typical' pairs of commuting operators can be simultaneously embedded into commuting pairs of C_{0}-semigroups, which extends results of Eisner (2009--10).

We introduce the FRactional-Order graph Neural Dynamical network (FROND), a new continuous graph neural network (GNN) framework. Unlike traditional continuous GNNs that rely on integer-order differential equations, FROND employs the Caputo fractional derivative to leverage the non-local properties of fractional calculus. This approach enables the capture of long-term dependencies in feature updates, moving beyond the Markovian update mechanisms in conventional integer-order models and offering enhanced capabilities in graph representation learning. We offer an interpretation of the node feature updating process in FROND from a non-Markovian random walk perspective when the feature updating is particularly governed by a diffusion process. We demonstrate analytically that oversmoothing can be mitigated in this setting. Experimentally, we validate the FROND framework by comparing the fractional adaptations of various established integer-order continuous GNNs, demonstrating their consistently improved performance and underscoring the framework's potential as an effective extension to enhance traditional continuous GNNs. The code is available at https://github.com/zknus/ICLR2024-FROND.

The space of unitary C_{0}-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of R_{+}, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [Eisner, 2010]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive C_{0}-semigroups constitute Polish spaces and thus positively addresses the open problem.

A central problem in learning from sequential data is representing cumulative history in an incremental fashion as more data is processed. We introduce a general framework (HiPPO) for the online compression of continuous signals and discrete time series by projection onto polynomial bases. Given a measure that specifies the importance of each time step in the past, HiPPO produces an optimal solution to a natural online function approximation problem. As special cases, our framework yields a short derivation of the recent Legendre Memory Unit (LMU) from first principles, and generalizes the ubiquitous gating mechanism of recurrent neural networks such as GRUs. This formal framework yields a new memory update mechanism (HiPPO-LegS) that scales through time to remember all history, avoiding priors on the timescale. HiPPO-LegS enjoys the theoretical benefits of timescale robustness, fast updates, and bounded gradients. By incorporating the memory dynamics into recurrent neural networks, HiPPO RNNs can empirically capture complex temporal dependencies. On the benchmark permuted MNIST dataset, HiPPO-LegS sets a new state-of-the-art accuracy of 98.3%. Finally, on a novel trajectory classification task testing robustness to out-of-distribution timescales and missing data, HiPPO-LegS outperforms RNN and neural ODE baselines by 25-40% accuracy.

Continuous-time generative models, such as diffusion models, flow matching, and rectified flow, learn time-dependent vector fields but are typically trained with objectives that treat timesteps independently, leading to high estimator variance and inefficient sampling. Prior approaches mitigate this via explicit smoothness penalties, trajectory regularization, or modified probability paths and solvers. We introduce Temporal Pair Consistency (TPC), a lightweight variance-reduction principle that couples velocity predictions at paired timesteps along the same probability path, operating entirely at the estimator level without modifying the model architecture, probability path, or solver. We provide a theoretical analysis showing that TPC induces a quadratic, trajectory-coupled regularization that provably reduces gradient variance while preserving the underlying flow-matching objective. Instantiated within flow matching, TPC improves sample quality and efficiency across CIFAR-10 and ImageNet at multiple resolutions, achieving lower FID at identical or lower computational cost than prior methods, and extends seamlessly to modern SOTA-style pipelines with noise-augmented training, score-based denoising, and rectified flow.

With the emergence of powerful representations of continuous data in the form of neural fields, there is a need for discretization invariant learning: an approach for learning maps between functions on continuous domains without being sensitive to how the function is sampled. We present a new framework for understanding and designing discretization invariant neural networks (DI-Nets), which generalizes many discrete networks such as convolutional neural networks as well as continuous networks such as neural operators. Our analysis establishes upper bounds on the deviation in model outputs under different finite discretizations, and highlights the central role of point set discrepancy in characterizing such bounds. This insight leads to the design of a family of neural networks driven by numerical integration via quasi-Monte Carlo sampling with discretizations of low discrepancy. We prove by construction that DI-Nets universally approximate a large class of maps between integrable function spaces, and show that discretization invariance also describes backpropagation through such models. Applied to neural fields, convolutional DI-Nets can learn to classify and segment visual data under various discretizations, and sometimes generalize to new types of discretizations at test time. Code: https://github.com/clintonjwang/DI-net.

An agent holds a position in a perpetual contract with payoff function ψ and attempts to liquidate the position while managing transaction costs, inventory risk, and funding rate payments. By solving the agent's stochastic control problem we obtain a closed-form expression for the optimal trading strategy when the payoff function is given by ψ(s) = s. When the payoff function is non-linear we provide approximations to the optimal strategy which apply when the funding rate parameter is small or when the length of the trading interval is small. We further prove that when ψ is non-linear, the short time approximation can be written in terms of the closed-form trading strategy corresponding to the case of the identity payoff function.

A fairly reliable trend in deep reinforcement learning is that the performance scales with the number of parameters, provided a complimentary scaling in amount of training data. As the appetite for large models increases, it is imperative to address, sooner than later, the potential problem of running out of high-quality demonstrations. In this case, instead of collecting only new data via costly human demonstrations or risking a simulation-to-real transfer with uncertain effects, it would be beneficial to leverage vast amounts of readily-available low-quality data. Since classical control algorithms such as behavior cloning or temporal difference learning cannot be used on reward-free or action-free data out-of-the-box, this solution warrants novel training paradigms for continuous control. We propose a simple algorithm called Diffused Value Function (DVF), which learns a joint multi-step model of the environment-robot interaction dynamics using a diffusion model. This model can be efficiently learned from state sequences (i.e., without access to reward functions nor actions), and subsequently used to estimate the value of each action out-of-the-box. We show how DVF can be used to efficiently capture the state visitation measure for multiple controllers, and show promising qualitative and quantitative results on challenging robotics benchmarks.

We consider the problem of approximating a function from L^2 by an element of a given m-dimensional space V_m, associated with some feature map varphi, using evaluations of the function at random points x_1,dots,x_n. After recalling some results on optimal weighted least-squares using independent and identically distributed points, we consider weighted least-squares using projection determinantal point processes (DPP) or volume sampling. These distributions introduce dependence between the points that promotes diversity in the selected features varphi(x_i). We first provide a generalized version of volume-rescaled sampling yielding quasi-optimality results in expectation with a number of samples n = O(mlog(m)), that means that the expected L^2 error is bounded by a constant times the best approximation error in L^2. Also, further assuming that the function is in some normed vector space H continuously embedded in L^2, we further prove that the approximation is almost surely bounded by the best approximation error measured in the H-norm. This includes the cases of functions from L^infty or reproducing kernel Hilbert spaces. Finally, we present an alternative strategy consisting in using independent repetitions of projection DPP (or volume sampling), yielding similar error bounds as with i.i.d. or volume sampling, but in practice with a much lower number of samples. Numerical experiments illustrate the performance of the different strategies.

Physics-Informed Neural Networks (PINNs) provide a mesh-free approach for solving differential equations by embedding physical constraints into neural network training. However, PINNs tend to overfit within the training domain, leading to poor generalization when extrapolating beyond trained spatiotemporal regions. This work presents SPIKE (Sparse Physics-Informed Koopman-Enhanced), a framework that regularizes PINNs with continuous-time Koopman operators to learn parsimonious dynamics representations. By enforcing linear dynamics dz/dt = Az in a learned observable space, both PIKE (without explicit sparsity) and SPIKE (with L1 regularization on A) learn sparse generator matrices, embodying the parsimony principle that complex dynamics admit low-dimensional structure. Experiments across parabolic, hyperbolic, dispersive, and stiff PDEs, including fluid dynamics (Navier-Stokes) and chaotic ODEs (Lorenz), demonstrate consistent improvements in temporal extrapolation, spatial generalization, and long-term prediction accuracy. The continuous-time formulation with matrix exponential integration provides unconditional stability for stiff systems while avoiding diagonal dominance issues inherent in discrete-time Koopman operators.

We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for optimization updates. We provide continuous-time analysis of CGO and its convergence properties while showing that in the continuous limit, CGO predecessors degenerate to their gradient descent ascent (GDA) variants. We provide a rate of convergence to stationary points and further propose a generalized class of alpha-coherent function for which we provide convergence analysis. We show that for strictly alpha-coherent functions, our algorithm convergences to a saddle point. Moreover, we propose optimistic CGO (OCGO), an optimistic variant, for which we show convergence rate to saddle points in alpha-coherent class of functions.

A temporal point process is a mathematical model for a time series of discrete events, which covers various applications. Recently, recurrent neural network (RNN) based models have been developed for point processes and have been found effective. RNN based models usually assume a specific functional form for the time course of the intensity function of a point process (e.g., exponentially decreasing or increasing with the time since the most recent event). However, such an assumption can restrict the expressive power of the model. We herein propose a novel RNN based model in which the time course of the intensity function is represented in a general manner. In our approach, we first model the integral of the intensity function using a feedforward neural network and then obtain the intensity function as its derivative. This approach enables us to both obtain a flexible model of the intensity function and exactly evaluate the log-likelihood function, which contains the integral of the intensity function, without any numerical approximations. Our model achieves competitive or superior performances compared to the previous state-of-the-art methods for both synthetic and real datasets.

Despite their spectacular progress, language models still struggle on complex reasoning tasks, such as advanced mathematics. We consider a long-standing open problem in mathematics: discovering a Lyapunov function that ensures the global stability of a dynamical system. This problem has no known general solution, and algorithmic solvers only exist for some small polynomial systems. We propose a new method for generating synthetic training samples from random solutions, and show that sequence-to-sequence transformers trained on such datasets perform better than algorithmic solvers and humans on polynomial systems, and can discover new Lyapunov functions for non-polynomial systems.

Recent work has shown that deep neural networks are capable of approximating both value functions and policies in reinforcement learning domains featuring continuous state and action spaces. However, to the best of our knowledge no previous work has succeeded at using deep neural networks in structured (parameterized) continuous action spaces. To fill this gap, this paper focuses on learning within the domain of simulated RoboCup soccer, which features a small set of discrete action types, each of which is parameterized with continuous variables. The best learned agent can score goals more reliably than the 2012 RoboCup champion agent. As such, this paper represents a successful extension of deep reinforcement learning to the class of parameterized action space MDPs.

From the Bayesian perspective, the category of conditional probabilities (a variant of the Kleisli category of the Giry monad, whose objects are measurable spaces and arrows are Markov kernels) gives a nice framework for conceptualization and analysis of many aspects of machine learning. Using categorical methods, we construct models for parametric and nonparametric Bayesian reasoning on function spaces, thus providing a basis for the supervised learning problem. In particular, stochastic processes are arrows to these function spaces which serve as prior probabilities. The resulting inference maps can often be analytically constructed in this symmetric monoidal weakly closed category. We also show how to view general stochastic processes using functor categories and demonstrate the Kalman filter as an archetype for the hidden Markov model.

Invariance learning methods aim to learn invariant features in the hope that they generalize under distributional shifts. Although many tasks are naturally characterized by continuous domains, current invariance learning techniques generally assume categorically indexed domains. For example, auto-scaling in cloud computing often needs a CPU utilization prediction model that generalizes across different times (e.g., time of a day and date of a year), where `time' is a continuous domain index. In this paper, we start by theoretically showing that existing invariance learning methods can fail for continuous domain problems. Specifically, the naive solution of splitting continuous domains into discrete ones ignores the underlying relationship among domains, and therefore potentially leads to suboptimal performance. To address this challenge, we then propose Continuous Invariance Learning (CIL), which extracts invariant features across continuously indexed domains. CIL is a novel adversarial procedure that measures and controls the conditional independence between the labels and continuous domain indices given the extracted features. Our theoretical analysis demonstrates the superiority of CIL over existing invariance learning methods. Empirical results on both synthetic and real-world datasets (including data collected from production systems) show that CIL consistently outperforms strong baselines among all the tasks.

Despite advances in learning-based methods, finding valid Lyapunov functions for nonlinear dynamical systems remains challenging. Current neural network approaches face two main issues: challenges in scalable verification and limited interpretability. To address these, we propose an end-to-end framework using transformers to construct analytical Lyapunov functions (local), which simplifies formal verification, enhances interpretability, and provides valuable insights for control engineers. Our framework consists of a transformer-based trainer that generates candidate Lyapunov functions and a falsifier that verifies candidate expressions and refines the model via risk-seeking policy gradient. Unlike Alfarano et al. (2024), which utilizes pre-training and seeks global Lyapunov functions for low-dimensional systems, our model is trained from scratch via reinforcement learning (RL) and succeeds in finding local Lyapunov functions for high-dimensional and non-polynomial systems. Given the analytical nature of the candidates, we employ efficient optimization methods for falsification during training and formal verification tools for the final verification. We demonstrate the efficiency of our approach on a range of nonlinear dynamical systems with up to ten dimensions and show that it can discover Lyapunov functions not previously identified in the control literature.

We consider characterisations of unitary dilations and approximations of irreversible classical dynamical systems on a Hilbert space. In the commutative case, building on the work in [9], one can express well known approximants (e.g. Hille- and Yosida-approximants) via expectations over certain stochastic processes. Using this, our first result characterises the simultaneous regular unitary dilatability of commuting families of C_{0}-semigroups via the dilatability of such approximants as well as via regular polynomial bounds. This extends the results in [13] to the unbounded setting. We secondly consider characterisations of unitary and regular unitary dilations via two distinct functional calculi. Applying these tools to a large class of classical dynamical systems, these two notions of dilation exactly characterise when a system admits unitary approximations under certain distinct notions of weak convergence. This establishes a sharp topological distinction between the two notions of unitary dilations. Our results are applicable to commutative systems as well as non-commutative systems satisfying the canonical commutation relations (CCR) in the Weyl form.

Residual neural networks are state-of-the-art deep learning models. Their continuous-depth analog, neural ordinary differential equations (ODEs), are also widely used. Despite their success, the link between the discrete and continuous models still lacks a solid mathematical foundation. In this article, we take a step in this direction by establishing an implicit regularization of deep residual networks towards neural ODEs, for nonlinear networks trained with gradient flow. We prove that if the network is initialized as a discretization of a neural ODE, then such a discretization holds throughout training. Our results are valid for a finite training time, and also as the training time tends to infinity provided that the network satisfies a Polyak-Lojasiewicz condition. Importantly, this condition holds for a family of residual networks where the residuals are two-layer perceptrons with an overparameterization in width that is only linear, and implies the convergence of gradient flow to a global minimum. Numerical experiments illustrate our results.

Recurrent neural networks (RNNs), temporal convolutions, and neural differential equations (NDEs) are popular families of deep learning models for time-series data, each with unique strengths and tradeoffs in modeling power and computational efficiency. We introduce a simple sequence model inspired by control systems that generalizes these approaches while addressing their shortcomings. The Linear State-Space Layer (LSSL) maps a sequence u mapsto y by simply simulating a linear continuous-time state-space representation x = Ax + Bu, y = Cx + Du. Theoretically, we show that LSSL models are closely related to the three aforementioned families of models and inherit their strengths. For example, they generalize convolutions to continuous-time, explain common RNN heuristics, and share features of NDEs such as time-scale adaptation. We then incorporate and generalize recent theory on continuous-time memorization to introduce a trainable subset of structured matrices A that endow LSSLs with long-range memory. Empirically, stacking LSSL layers into a simple deep neural network obtains state-of-the-art results across time series benchmarks for long dependencies in sequential image classification, real-world healthcare regression tasks, and speech. On a difficult speech classification task with length-16000 sequences, LSSL outperforms prior approaches by 24 accuracy points, and even outperforms baselines that use hand-crafted features on 100x shorter sequences.

Functional data play a pivotal role across science and engineering, yet their infinite-dimensional nature makes representation learning challenging. Conventional statistical models depend on pre-chosen basis expansions or kernels, limiting the flexibility of data-driven discovery, while many deep-learning pipelines treat functions as fixed-grid vectors, ignoring inherent continuity. In this paper, we introduce Functional Attention with a Mixture-of-Experts (FAME), an end-to-end, fully data-driven framework for function-on-function regression. FAME forms continuous attention by coupling a bidirectional neural controlled differential equation with MoE-driven vector fields to capture intra-functional continuity, and further fuses change to inter-functional dependencies via multi-head cross attention. Extensive experiments on synthetic and real-world functional-regression benchmarks show that FAME achieves state-of-the-art accuracy, strong robustness to arbitrarily sampled discrete observations of functions.

Spiking Neural Networks (SNNs) can achieve competitive performance by converting already existing well-trained Artificial Neural Networks (ANNs), avoiding further costly training. This property is particularly attractive in Reinforcement Learning (RL), where training through environment interaction is expensive and potentially unsafe. However, existing conversion methods perform poorly in continuous control, where suitable baselines are largely absent. We identify error amplification as the key cause: small action approximation errors become temporally correlated across decision steps, inducing cumulative state distribution shift and severe performance degradation. To address this issue, we propose Cross-Step Residual Potential Initialization (CRPI), a lightweight gradient-free mechanism that carries over residual membrane potentials across decision steps to suppress temporally correlated errors. Experiments on continuous control benchmarks with both vector and visual observations demonstrate that CRPI can be integrated into existing conversion pipelines and substantially recovers lost performance. Our results highlight continuous control as a critical and challenging benchmark for ANN-to-SNN conversion, where small errors can be strongly amplified and impact performance. Code is available at https://github.com/xuzijie32/ANN2SNN-CRPI.

This work explores the in-context learning capabilities of State Space Models (SSMs) and presents, to the best of our knowledge, the first theoretical explanation of a possible underlying mechanism. We introduce a novel weight construction for SSMs, enabling them to predict the next state of any dynamical system after observing previous states without parameter fine-tuning. This is accomplished by extending the HiPPO framework to demonstrate that continuous SSMs can approximate the derivative of any input signal. Specifically, we find an explicit weight construction for continuous SSMs and provide an asymptotic error bound on the derivative approximation. The discretization of this continuous SSM subsequently yields a discrete SSM that predicts the next state. Finally, we demonstrate the effectiveness of our parameterization empirically. This work should be an initial step toward understanding how sequence models based on SSMs learn in context.

Normalization layers and activation functions are fundamental components in deep networks and typically co-locate with each other. Here we propose to design them using an automated approach. Instead of designing them separately, we unify them into a single tensor-to-tensor computation graph, and evolve its structure starting from basic mathematical functions. Examples of such mathematical functions are addition, multiplication and statistical moments. The use of low-level mathematical functions, in contrast to the use of high-level modules in mainstream NAS, leads to a highly sparse and large search space which can be challenging for search methods. To address the challenge, we develop efficient rejection protocols to quickly filter out candidate layers that do not work well. We also use multi-objective evolution to optimize each layer's performance across many architectures to prevent overfitting. Our method leads to the discovery of EvoNorms, a set of new normalization-activation layers with novel, and sometimes surprising structures that go beyond existing design patterns. For example, some EvoNorms do not assume that normalization and activation functions must be applied sequentially, nor need to center the feature maps, nor require explicit activation functions. Our experiments show that EvoNorms work well on image classification models including ResNets, MobileNets and EfficientNets but also transfer well to Mask R-CNN with FPN/SpineNet for instance segmentation and to BigGAN for image synthesis, outperforming BatchNorm and GroupNorm based layers in many cases.

We consider offline reinforcement learning (RL) methods in possibly nonstationary environments. Many existing RL algorithms in the literature rely on the stationarity assumption that requires the system transition and the reward function to be constant over time. However, the stationarity assumption is restrictive in practice and is likely to be violated in a number of applications, including traffic signal control, robotics and mobile health. In this paper, we develop a consistent procedure to test the nonstationarity of the optimal Q-function based on pre-collected historical data, without additional online data collection. Based on the proposed test, we further develop a sequential change point detection method that can be naturally coupled with existing state-of-the-art RL methods for policy optimization in nonstationary environments. The usefulness of our method is illustrated by theoretical results, simulation studies, and a real data example from the 2018 Intern Health Study. A Python implementation of the proposed procedure is available at https://github.com/limengbinggz/CUSUM-RL.

Learning physical simulations has been an essential and central aspect of many recent research efforts in machine learning, particularly for Navier-Stokes-based fluid mechanics. Classic numerical solvers have traditionally been computationally expensive and challenging to use in inverse problems, whereas Neural solvers aim to address both concerns through machine learning. We propose a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluate a large set of basis functions in the context of (a) a compressible 1D SPH simulation, (b) a weakly compressible 2D SPH simulation, and (c) an incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries included in the basis functions are key aspects of stability and accuracy. Our broad evaluation shows that Fourier-based continuous convolutions outperform all other architectures regarding accuracy and generalization. Finally, using these Fourier-based networks, we show that prior inductive biases, such as window functions, are no longer necessary. An implementation of our approach, as well as complete datasets and solver implementations, is available at https://github.com/tum-pbs/SFBC.

We present metrics for measuring state similarity in Markov decision processes (MDPs) with infinitely many states, including MDPs with continuous state spaces. Such metrics provide a stable quantitative analogue of the notion of bisimulation for MDPs, and are suitable for use in MDP approximation. We show that the optimal value function associated with a discounted infinite horizon planning task varies continuously with respect to our metric distances.

In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a distributed universal approximation theorem guaranteeing that any Lipschitz non-linear operator between L^2([0,1]^d) spaces can be approximated uniformly over the Sobolev unit ball therein, to any given varepsilon>0 accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of O(varepsilon^{-1}). Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies varepsilon. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of L^2([0,1]^d).

The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the theory to the study of stochastic dynamical systems in discrete-time, and give a brief exposition of the Wiener process as a foundation for stochastic differential equations. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.

Diffusion models for continuous data gained widespread adoption owing to their high quality generation and control mechanisms. However, controllable diffusion on discrete data faces challenges given that continuous guidance methods do not directly apply to discrete diffusion. Here, we provide a straightforward derivation of classifier-free and classifier-based guidance for discrete diffusion, as well as a new class of diffusion models that leverage uniform noise and that are more guidable because they can continuously edit their outputs. We improve the quality of these models with a novel continuous-time variational lower bound that yields state-of-the-art performance, especially in settings involving guidance or fast generation. Empirically, we demonstrate that our guidance mechanisms combined with uniform noise diffusion improve controllable generation relative to autoregressive and diffusion baselines on several discrete data domains, including genomic sequences, small molecule design, and discretized image generation.

Neural networks have proven to be a highly effective tool for solving complex problems in many areas of life. Recently, their importance and practical usability have further been reinforced with the advent of deep learning. One of the important conditions for the success of neural networks is the choice of an appropriate activation function introducing non-linearity into the model. Many types of these functions have been proposed in the literature in the past, but there is no single comprehensive source containing their exhaustive overview. The absence of this overview, even in our experience, leads to redundancy and the unintentional rediscovery of already existing activation functions. To bridge this gap, our paper presents an extensive survey involving 400 activation functions, which is several times larger in scale than previous surveys. Our comprehensive compilation also references these surveys; however, its main goal is to provide the most comprehensive overview and systematization of previously published activation functions with links to their original sources. The secondary aim is to update the current understanding of this family of functions.

In this article, we present a structured Kalman filter associated with the transformation matrix for observable Kalman canonical decomposition from conventional Kalman filter (CKF) in order to generate a more accurate time scale. The conventional Kalman filter is a special case of the proposed structured Kalman filter which yields the same predicted unobservable or observable states when some conditions are satisfied. We consider an optimization problem respective to the transformation matrix where the objective function is associated with not only the expected value of prediction error but also its variance. We reveal that such an objective function is a convex function and show some conditions under which CKF is nothing but the optimal algorithm if ideal computation is possible without computation error. A numerical example is presented to show the robustness of the proposed method in terms of the initial error covariance

We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps E_{τ,f} built from a Markov kernel, a coarse-graining lens, and a time scale τ. We introduce a computable total-variation idempotence defect for E_{τ,f}; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen τ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.

The deep learning recipe of casting real-world problems as mathematical optimisation and tackling the optimisation by training deep neural networks using gradient-based optimisation has undoubtedly proven to be a fruitful one. The understanding behind why deep learning works, however, has lagged behind its practical significance. We aim to make steps towards an improved understanding of deep learning with a focus on optimisation and model regularisation. We start by investigating gradient descent (GD), a discrete-time algorithm at the basis of most popular deep learning optimisation algorithms. Understanding the dynamics of GD has been hindered by the presence of discretisation drift, the numerical integration error between GD and its often studied continuous-time counterpart, the negative gradient flow (NGF). To add to the toolkit available to study GD, we derive novel continuous-time flows that account for discretisation drift. Unlike the NGF, these new flows can be used to describe learning rate specific behaviours of GD, such as training instabilities observed in supervised learning and two-player games. We then translate insights from continuous time into mitigation strategies for unstable GD dynamics, by constructing novel learning rate schedules and regularisers that do not require additional hyperparameters. Like optimisation, smoothness regularisation is another pillar of deep learning's success with wide use in supervised learning and generative modelling. Despite their individual significance, the interactions between smoothness regularisation and optimisation have yet to be explored. We find that smoothness regularisation affects optimisation across multiple deep learning domains, and that incorporating smoothness regularisation in reinforcement learning leads to a performance boost that can be recovered using adaptions to optimisation methods.

The optimization of the latents and parameters of diffusion models with respect to some differentiable metric defined on the output of the model is a challenging and complex problem. The sampling for diffusion models is done by solving either the probability flow ODE or diffusion SDE wherein a neural network approximates the score function allowing a numerical ODE/SDE solver to be used. However, naive backpropagation techniques are memory intensive, requiring the storage of all intermediate states, and face additional complexity in handling the injected noise from the diffusion term of the diffusion SDE. We propose a novel family of bespoke ODE solvers to the continuous adjoint equations for diffusion models, which we call AdjointDEIS. We exploit the unique construction of diffusion SDEs to further simplify the formulation of the continuous adjoint equations using exponential integrators. Moreover, we provide convergence order guarantees for our bespoke solvers. Significantly, we show that continuous adjoint equations for diffusion SDEs actually simplify to a simple ODE. Lastly, we demonstrate the effectiveness of AdjointDEIS for guided generation with an adversarial attack in the form of the face morphing problem. Our code will be released on our project page https://zblasingame.github.io/AdjointDEIS/

Can we build continuous generative models which generalize across scales, can be evaluated at any coordinate, admit calculation of exact derivatives, and are conceptually simple? Existing MLP-based architectures generate worse samples than the grid-based generators with favorable convolutional inductive biases. Models that focus on generating images at different scales do better, but employ complex architectures not designed for continuous evaluation of images and derivatives. We take a signal-processing perspective and treat continuous image generation as interpolation from samples. Indeed, correctly sampled discrete images contain all information about the low spatial frequencies. The question is then how to extrapolate the spectrum in a data-driven way while meeting the above design criteria. Our answer is FunkNN -- a new convolutional network which learns how to reconstruct continuous images at arbitrary coordinates and can be applied to any image dataset. Combined with a discrete generative model it becomes a functional generator which can act as a prior in continuous ill-posed inverse problems. We show that FunkNN generates high-quality continuous images and exhibits strong out-of-distribution performance thanks to its patch-based design. We further showcase its performance in several stylized inverse problems with exact spatial derivatives.

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive C_{0}-semigroups under the topology of uniform weak operator convergence on compact subsets of R_{+}. Eisner and Serény raised in 2009 the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The transfer result in turn relies upon classification of topological properties via infinite games. Our approach is sufficiently general and can be applied to other contexts, e.g. the space of contractions under the pw-topology.

Standard continuous-time generative models rely on monolithic architectures that must navigate vastly different signal regimes, from isotropic noise to intricate data distributions. While scaling model capacity improves performance, deploying a massive network uniformly across the entire generative timeline is inherently inefficient. In this work, we propose Complexity-Balanced Splitting (CBS), a principled framework for temporal capacity allocation that distributes the generative workload across multiple specialized sub-networks. Grounded in function approximation theory and de Boor's equidistribution principle, CBS partitions the diffusion timeline into segments of equal approximation burden, allocating more representational capacity to regions where the generative dynamics are more difficult to model. To estimate this local complexity, we introduce two complementary and tractable monitor functions: a spatial measure based on the flow's Dirichlet energy, and a geometric measure based on the acceleration of the sampling trajectories. Using a lightweight auxiliary model to estimate these complexity profiles, our approach eliminates the need for heuristic temporal splits or computationally expensive search procedures. Extensive evaluation across multiple architectures (SiT, JiT, and UNet) and datasets demonstrates that CBS consistently improves synthesis quality without increasing per-step inference cost. In particular, CBS improves FID by ~35% on SiT-XL with CFG relative to naive temporal partitioning. Project page is available at https://noamissachar.github.io/CBS/.

Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

Pairwise dependence measures such as correlation and causality are fundamental to temporal data mining, yet there is still no principled and robust way to quantify dependence between heterogeneous data types, especially between continuous time series and discrete temporal event sequences. Existing approaches rely on ad hoc transformations or mutual-information estimators that are highly sensitive to quantization, repeated values, and event redundancy, leading to biased or unstable results in practice. We propose a nonparametric mutual information estimator that directly measures the dependence between time series and event sequences without data transformation, learning, or ad hoc discretization. Our method models the continuous-discrete duality of real-world time series to handle quantization and repeated-value artifacts and introduces a latent event clustering strategy to mitigate bias from event co-occurrence and redundancy. Together, these yield a robust and unified framework that bridges discrete and continuous mutual information. We evaluate the proposed estimator on four representative tasks: discrete-continuous time-delayed mutual information for causality analysis, global and local temporal repetition discovery, discrete covariate selection for time series forecasting, and continuous feature selection for classification. Experiments on synthetic and real-world datasets show consistent improvements over existing methods in accuracy, robustness, and interpretability, positioning our approach as a general-purpose dependence operator for heterogeneous temporal data, similar to Pearson correlation for homogeneous time series. Code available at: https://github.com/HaojiHu/Multimodal-Temporal-Data-Quantification