Stochastic Differential Equations

Also known as SDE

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Latest papers 101

21 papers in the last four weeks, up 320% on the four weeks before. 0.1% of all new papers.

May 14, 2026

Unbiased and Second-Order-Free Training for High-Dimensional PDEs

Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler-Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE.

Jaemin Seo, Surin Lee, Jae Yong Lee

Department of Artificial Intelligence, Chung-Ang University, Seoul, Republic of Korea.

Stochastic Differential Equations Neural PDE Solvers PDE Solving

May 11, 2026

Variational Inference for Lévy Process-Driven SDEs via Neural Tilting

Modelling extreme events and heavy-tailed phenomena is central to building reliable predictive systems in domains such as finance, climate science, and safety-critical AI. While Lévy processes provide a natural mathematical framework for capturing jumps and heavy tails, Bayesian inference for Lévy-driven stochastic differential equations (SDEs) remains intractable with existing methods: Monte Carlo approaches are rigorous but lack scalability, whereas neural variational inference methods are efficient but rely on Gaussian assumptions that fail to capture discontinuities. We address this tension by introducing a neural exponential tilting framework for variational inference in Lévy-driven SDEs. Our approach constructs a flexible variational family by exponentially reweighting the Lévy measure using neural networks. This parametrization preserves the jump structure of the underlying process while remaining computationally tractable. To enable efficient inference, we develop a quadratic neural parametrization that yields closed-form normalization of the tilted measure, a conditional Gaussian representation for stable processes that facilitates simulation, and symmetry-aware Monte Carlo estimators for scalable optimization. Empirically, we demonstrate that the method accurately captures jump dynamics and yields reliable posterior inference in regimes where Gaussian-based variational approaches fail, on both synthetic and real-world datasets.

Yaman Kindap, Manfred Opper, Benjamin Dupuis +2

1Imperial College London, UK · 2Technical University of Berlin, Germany · 3INRIA, CNRS, Département d’Informatique de l’Ecole Normale Supérieure / PSL, France

Stochastic Differential Equations Bayesian Inference Variational Inference

May 10, 2026

Learning stochastic multiscale models through normalizing flows

Many systems in physics, engineering, and biology exhibit multiscale stochastic dynamics, where low-dimensional slow variables evolve under the influence of high-dimensional fast processes. In practice, observations are often limited to a single trajectory of the slow component, while the fast dynamics remain unobserved, making statistical learning challenging. Approaches based on partial differential equations (PDE), such as Fokker-Planck formulations, aim to characterize the evolution of probability densities, typically requiring dense space-time data or grid-based solvers. In contrast, we adopt a trajectory-based perspective and develop a data-driven framework for learning effective stochastic dynamics from a single observed path. We model the dynamics by coupled multiscale stochastic differential equations (SDEs) and first obtain a principled model reduction through stochastic averaging. Unlike generic model reduction techniques such as PCA, this respects the dynamical structure of the original system and explicitly incorporates the interaction between slow and fast scales. A central challenge, however, is that the reduced model depends on the invariant distribution of the fast process, which is a solution to an intractable and often unknown PDE. We introduce a novel learning framework that parameterizes the invariant distribution using normalizing flows, enabling expressive density modeling in the latent fast-variable space. The flow is trained end-to-end by optimizing a penalized likelihood objective induced by the reduced stochastic dynamics. Furthermore, we develop a Bayesian variational inference procedure for uncertainty quantification, employing a second normalizing flow to approximate the posterior distribution over model parameters. This yields a scalable approach to capturing epistemic uncertainty in multiscale systems.

Anan Saha, Arnab Ganguly

Department of Mathematics Louisiana State University Baton Rouge, LA 70802

Stochastic Differential Equations Multiscale Time-Series Modeling

May 9, 2026

Deterministic Decomposition of Stochastic Generative Dynamics

Modern generative models can be understood as probability transport from a simple base distribution to a target data distribution. Deterministic transport models offer tractable velocity-field parameterizations, whereas stochastic generative models capture richer density evolution through drift and diffusion. Yet when stochastic dynamics are described through deterministic velocity fields, the effects of drift and diffusion are often compressed into a single effective field, obscuring the distinct roles of deterministic evolution and stochastic fluctuation. In this work, we show that the deterministic field

b_t

of a stochastic generative process admits a natural transport--osmotic decomposition that separates deterministic transport from stochastic, diffusion-induced effects:

b_t = u_t + d_t

, where

u_t

governs marginal probability transport and

d_t

captures an osmotic effect induced by diffusion and determined by the marginal score. Based on this decomposition, we propose Bridge Matching, a flow-based framework for learning decomposed generative dynamics through both marginal and conditional formulations. In generative modeling experiments, we recombine the learned components as

b_t = u_t + λ_d d_t

, showing that the proposed decomposition enables interpretable and controllable sampling by adjusting the osmotic contribution in probability transport.

Xingyu Song, Yuan Mei, Naoya Takeishi

1The University of Tokyo · 2Zhejiang University · *Visiting student at The University of Tokyo from Zhejiang University.

Flow Matching Stochastic Differential Equations Generative Modeling Flow-Based Generative Modeling

May 7, 2026

Diffusion-Based Posterior Sampling: A Feynman-Kac Analysis of Bias and Stability

Diffusion-based posterior samplers draw from measurement- or reward-conditioned posteriors using pretrained diffusion priors. Though widely used for inverse problems, their output is biased even with exact prior scores. First, we quantify this bias for the DPS algorithm, the base method for a family of samplers, by introducing a surrogate curve of probability densities which interpolates between the true posterior and the standard Gaussian. The ratio of sampler to surrogate densities solves a parabolic PDE, and its Feynman--Kac formula yields an explicit representation of this ratio as the expectation of a path functional. This representation reveals which posterior regions are over- or under-sampled. This framework also provides a design principle for bias reduction that includes STSL, a related sampler. Second, in hard-constraint problems (e.g. the low measurement noise regime) these samplers are known to be numerically unstable. We find that this instability of discretized DPS arises from an effective guidance update whose strength scales inversely with step size, inevitably violating the forward-Euler stability criterion near the constraint. This finding motivates implicit (backward-Euler) guidance updates. Finally, we show that the standard practice of early guidance-stopping, used to avoid oscillations, introduces a secondary bias which we characterize through our framework.

William Porteous, Advait Parulekar, Sebastien Motsch +2

Department of Mathematics, UT Austin. · Department of Electrical and Computer Engineering, UT Austin. · School of Mathematical and Statistical Sciences, Arizona State University.

Stochastic Differential Equations Bayesian Inverse Problems Diffusion Guidance Diffusion Sampling Diffusion Posterior Sampling

May 7, 2026

Variational Smoothing and Inference for SDEs from Sparse Data with Dynamic Neural Flows

Stochastic differential equations (SDEs) provide a flexible framework for modeling temporal dynamics in partially observed systems. A central task is to calibrate such models from data, which requires inferring latent trajectories and parameters from sparse, noisy observations. Classical smoothing methods for this problem are often limited by path degeneracy and poor scalability. In this work, we developed a novel method based on characterization of the posterior SDE in terms of conditional backward-in-time score defined as the gradient of a function solving a Kolmogorov backward equation with multiplicative updates at observation times. We learn this conditional score using neural networks trained to satisfy both the governing PDE and the observation-induced jump conditions, thereby integrating continuous-time dynamics with discrete Bayesian updates. The resulting score induces a posterior SDE with the same diffusion coefficient but a modified drift, enabling efficient posterior trajectory sampling. We further derive a likelihood-based objective for learning the SDE parameters, yielding an evidence lower bound (ELBO) for joint state smoothing and parameter estimation. This leads to a variational EM-style procedure, where the neural conditional score is optimized to approximate the smoothing distribution, followed by a maximization step over the SDE parameters using samples from the induced posterior. Experiments on nonlinear systems demonstrate accurate and stable inference with a very few observations demonstrating significant improved scalability compared to classical MCMC methods.

Yu Wang, Arnab Ganguly

Department of Mathematics Louisiana State University

Stochastic Differential Equations Bayesian Inference Variational Inference

May 2, 2026

Unifying Deep Stochastic Processes for Image Enhancement

Deep stochastic processes have recently become a central paradigm for image enhancement, with many methods explicitly conditioning the stochastic trajectory on the degraded input. However, the relationship between these conditional processes and standard diffusion models remains unclear. In this work, we introduce a unified perspective on stochastic image enhancement by classifying recent methods into three families of continuous-time processes: unconditional diffusion models, Ornstein-Uhlenbeck (OU) processes, and diffusion bridges. We show that all of these approaches arise from a common stochastic differential equation (SDE) formulation. This framework makes explicit that seemingly disparate methods differ primarily in their drift and diffusion terms, terminal distributions, and boundary conditions, while schedulers and samplers constitute orthogonal design choices. Leveraging this unification, we conduct a controlled empirical study across multiple image enhancement tasks using identical architectures and training protocols. Our results reveal no consistently dominant method; instead, we identify and disentangle the specific design choices that most strongly influence performance. Finally, we release ItoVision, a modular PyTorch library that implements the unified framework and enables rapid prototyping and fair comparison of stochastic image enhancement methods.

Wojciech Kozłowski, Radosław Kuczbański, Kamil Adamczewski +2

Wrocław University of Science and Technology, Wrocław, Poland · Tooploox, Warsaw, Poland

Image Enhancement Stochastic Differential Equations Diffusion Models

Apr 22, 2026

Structure-Aware Variational Learning of a Class of Generalized Diffusions

Learning the underlying potential energy of stochastic gradient systems from partial and noisy observations is a fundamental problem arising in physics, chemistry, and data-driven modeling. Classical approaches often rely on direct regression of governing equations or velocity fields, which can be sensitive to noise and external perturbations and may fail when observations are incomplete. In this work, we propose a structure-aware, energy-based learning framework for inferring unknown potential functions in generalized diffusion processes, grounded in the energetic variational approach. Starting from the energy-dissipation law associated with the Fokker-Planck equation, we construct loss functions based on the De Giorgi dissipation functional, which consistently couple the free energy and the dissipation mechanism of the system. This formulation avoids explicit enforcement of the governing partial differential equation and preserves the underlying variational structure of the dynamics. Through numerical experiments in one, two, and three dimensions, we demonstrate that the proposed energy-based loss exhibits enhanced robustness with respect to observation time, noise level, and the diversity and amount of available training data. These results highlight the effectiveness of energy-dissipation principles as a reliable foundation for learning stochastic diffusion dynamics from data.

Yubin Lu, Xiaofan Li, Chun Liu +2

School of Mathematics, South China University of Technology, Guangzhou 510640, China. · Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616 United States · School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332 United States +1

Stochastic Differential Equations

Apr 18, 2026

Extraction of informative statistical features in the problem of forecasting time series generated by It{ô}-type processes

In this paper, we consider the problem of extraction of most informative features from time series that are regarded as observed values of stochastic processes satisfying the It{ô} stochastic differential equations with unknown random drift and diffusion coefficients. We do not attract any additional information and use only the information contained in the time series as it is. Therefore, as additional features, we use the parameters of statistically adjusted mixture-type models of the observed regularities of the behavior of the time series. Several algorithms of construction of these parameters are discussed. These algorithms are based on statistical reconstruction of the coefficients which, in turn, is based on statistical separation of normal mixtures. We obtain two types of parameters by the techniques of the uniform and non-uniform statistical reconstruction of the coefficients of the underlying It{ô} process. The reconstructed coefficients obtained by uniform techniques do not depend on the current value of the process, while the non-uniform techniques reconstruct the coefficients with the account of their dependence on the value of the process. Actually, the non-uniform techniques used in this paper represent a stochastic analog of the Taylor expansion for the time series. The efficiency of the obtained additional features is compared by using them in the autoregressive algorithms of prediction of time series. In order to obtain pure conclusion that is not affected by unwanted factors, say, related to a special choice of the architecture of the neural network prediction methods, we used only simple autoregressive algorithms. We show that the use of additional statistical features improves the prediction.

Victor Korolev, Mikhail Ivanov, Tatiana Kukanova +4

Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics · Moscow Center for Fundamental and Applied Mathematics · Federal Research Center “Computer Science and Control”, Russian Academy of Sciences +2

Stochastic Differential Equations Time Series Forecasting

Apr 8, 2026

Diffusion Processes on Implicit Manifolds

High-dimensional data are often assumed to lie on lower-dimensional manifolds. We study how to construct diffusion processes on this data manifold using only point cloud samples and without access to charts, projections, or other geometric primitives. Here, we introduce Implicit Manifold-valued Diffusions (IMDs), a data-driven mathematical formalism for defining stochastic differential equations in the original high-dimensional space that describe drifting Brownian particles evolving intrinsically on the underlying manifold. Our construction hinges on approximating the corresponding infinitesimal generator of the diffusion process using a proximity graph over the data and using the carré-du-champ of the generator, which encodes the local tangent spaces of the manifold and lifts the intrinsic process into ambient coordinates. We show that as the number of samples grows, our discrete diffusion process converges in law on the space of probability paths to its smooth manifold counterpart. We further present an Euler-Maruyama scheme for the numerical integration of IMDs. We validate our framework using numerical experiments on synthetic manifolds and the MNIST data manifold, showing that IMDs remain confined over the manifold and enable its guided exploration. Our work provides the mathematical foundation and practical implementations of diffusion processes on data manifolds, opening new avenues for manifold-aware sampling, exploration, and generative modeling.

Victor Kawasaki-Borruat, Clara Grotehans, Pierre Vandergheynst +1

Signal Processing Laboratory 2 EPFL · Institute of Artificial Intelligence Medical University of Vienna

Stochastic Differential Equations Manifold Learning

Apr 8, 2026

Score Shocks: The Burgers Equation Structure of Diffusion Generative Models

We analyze the score field of a diffusion generative model through a Burgers-type evolution law. For VE diffusion, the heat-evolved data density implies that the score obeys viscous Burgers in one dimension and the corresponding irrotational vector Burgers system in

\R^d

, giving a PDE view of speciation transitions as the sharpening of inter-mode interfaces. For any binary decomposition of the noised density into two positive heat solutions, the score separates into a smooth background and a universal

\tanh

interfacial term determined by the component log-ratio; near a regular binary mode boundary this yields a normal criterion for speciation. In symmetric binary Gaussian mixtures, the criterion recovers the critical diffusion time detected by the midpoint derivative of the score and agrees with the spectral criterion of Biroli, Bonnaire, de~Bortoli, and Mézard (2024). After subtracting the background drift, the inter-mode layer has a local Burgers

\tanh

profile, which becomes global in the symmetric Gaussian case with width

σ_τ^2/a

. We also quantify exponential amplification of score errors across this layer, show that Burgers dynamics preserves irrotationality, and use a change of variables to reduce the VP-SDE to the VE case, yielding a closed-form VP speciation time. Gaussian-mixture formulas are verified to machine precision, and the local theorem is checked numerically on a quartic double-well.

Krisanu Sarkar

Indian Institute of Technology Bombay, Mumbai, India

Diffusion Models Score-Based Generative Modeling Stochastic Differential Equations

Apr 6, 2026

Generative Path-Law Jump-Diffusion: Sequential MMD-Gradient Flows and Generalisation Bounds in Marcus-Signature RKHS

This paper introduces a novel generative framework for synthesising forward-looking, càdlàg stochastic trajectories that are sequentially consistent with time-evolving path-law proxies, thereby incorporating anticipated structural breaks, regime shifts, and non-autonomous dynamics. By framing path synthesis as a sequential matching problem on restricted Skorokhod manifolds, we develop the Anticipatory Neural Jump-Diffusion (ANJD) flow, a generative mechanism that effectively inverts the time-extended Marcus-sense signature. Central to this approach is the Anticipatory Variance-Normalised Signature Geometry (AVNSG), a time-evolving precision operator that performs dynamic spectral whitening on the signature manifold to ensure contractivity during volatile regime shifts and discrete aleatoric shocks. We provide a rigorous theoretical analysis demonstrating that the joint generative flow constitutes an infinitesimal steepest descent direction for the Maximum Mean Discrepancy functional relative to a moving target proxy. Furthermore, we establish statistical generalisation bounds within the restricted path-space and analyse the Rademacher complexity of the whitened signature functionals to characterise the expressive power of the model under heavy-tailed innovations. The framework is implemented via a scalable numerical scheme involving Nyström-compressed score-matching and an anticipatory hybrid Euler-Maruyama-Marcus integration scheme. Our results demonstrate that the proposed method captures the non-commutative moments and high-order stochastic texture of complex, discontinuous path-laws with high computational efficiency.

Daniel Bloch

UNIVERSITY OF PARIS 6 & VINUNIVERSITY · Visting Professor at the College of Engineering and Computer Science, VinUniversity, Hanoi.

Flow-Based Generative Modeling Stochastic Differential Equations

Apr 1, 2026

Embedded Variational Neural Stochastic Differential Equations for Learning Heterogeneous Dynamics

This study examines the challenges of modeling complex and noisy data related to socioeconomic factors over time, with a focus on data from various districts in Odisha, India. Traditional time-series models struggle to capture both trends and variations together in this type of data. To tackle this, a Variational Neural Stochastic Differential Equation (V-NSDE) model is designed that combines the expressive dynamics of Neural SDEs with the generative capabilities of Variational Autoencoders (VAEs). This model uses an encoder and a decoder. The encoder takes the initial observations and district embeddings and translates them into a Gaussian distribution, which determines the mean and log-variance of the first latent state. Then the obtained latent state initiates the Neural SDE, which utilize neural networks to determine the drift and diffusion functions that govern continuous-time latent dynamics. These governing functions depend on the time index, latent state, and district embedding, which help the model learn the unique characteristics specific to each district. After that, using a probabilistic decoder, the observations are reconstructed from the latent trajectory. The decoder outputs a mean and log-variance for each time step, which follows the Gaussian likelihood. The Evidence Lower Bound (ELBO) training loss improves by adding a KL-divergence regularization term to the negative log-likelihood (nll). The obtained results demonstrate the effective learning of V-NSDE in recognizing complex patterns over time, yielding realistic outcomes that include clear trends and random fluctuations across different areas.

Sandeep Kumar Samota, Reema Gupta, Snehashish Chakraverty

Department of Mathematics, National Institute of Technology, Rourkela, India · Poverty Alleviation Research Centre, National Institute of Technology, Rourkela, India

Latent Dynamics Modeling Time Series Forecasting Stochastic Differential Equations Variational Autoencoders

Mar 31, 2026

Learning noisy phase transition dynamics from stochastic partial differential equations

The non-equilibrium dynamics of mesoscale phase transitions are fundamentally shaped by thermal fluctuations, which not only seed instabilities but actively control kinetic pathways, including rare barrier-crossing events such as nucleation that are entirely inaccessible to deterministic models. Machine-learning surrogates for such systems must therefore represent stochasticity explicitly, enforce conservation laws by construction, and expose physically interpretable structure. We develop physics-aware surrogate models for the stochastic Cahn-Hilliard equation in 3D that satisfy all three requirements simultaneously. The key innovation is to parameterize the surrogate at the level of inter-cell fluxes, decomposing each flux into a deterministic mobility-weighted chemical-potential gradient and a learnable noise amplitude. This design guarantees exact mass conservation at every step and adds physical fluctuations to inter-cell mass transport. A learnable free energy functional provides thermodynamic interpretability, validated by independent recovery of the bulk double-well landscape, interfacial excess energy, and curvature-independent interfacial tension. Tests demonstrate accurate reproduction of ensemble statistics and noise-accelerated coarsening, with generalization to spatial domains 64 times larger in volume and temporal horizons 160x longer than those seen during training. Critically, the stochastic surrogate captures thermally activated nucleation in the metastable regime, a qualitative capability that no deterministic surrogate can provide regardless of training, thus establishing flux-level stochasticity as an architectural necessity rather than an optional enhancement.

Luning Sun, Van Hai Nguyen, Shusen Liu +2

Lawrence Livermore National Laboratory, Livermore, CA, 94550,USA · The University of Texas at Austin, Austin, TX, 78712, USA

PDE Surrogate Modeling Phase Transitions Stochastic Differential Equations

Mar 27, 2026

Parameter Estimation in Stochastic Differential Equations via Wiener Chaos Expansion and Stochastic Gradient Descent

This study addresses the inverse problem of parameter estimation for Stochastic Differential Equations (SDEs) by minimizing a regularized discrepancy functional via Stochastic Gradient Descent (SGD). To achieve computational efficiency, we leverage the Wiener Chaos Expansion (WCE), a spectral decomposition technique that projects the stochastic solution onto an orthogonal basis of Hermite polynomials. This transformation effectively maps the stochastic dynamics into a hierarchical system of deterministic functions, termed the propagator. By reducing the stochastic inference task to a deterministic optimization problem, our framework circumvents the heavy computational burden and sampling requirements of traditional simulation-based methods like MCMC or MLE. The robustness and scalability of the proposed approach are demonstrated through numerical experiments on various non-linear SDEs, including models for individual biological growth. Results show that the WCE-SGD framework provides accurate parameter recovery even from discrete, noisy observations, offering a significant paradigm shift in the efficient modeling of complex stochastic systems.

Francisco Delgado-Vences, José Julián Pavón-Español, Arelly Ornelas

Universitat Autónoma de Barcelona, Departament de Matemàtiques, Edifici C, Facultat de Ciències, 080193, Bellaterra, Spain · Facultad de Ciencias - Universidad Nacional Autónoma de México. Ciudad Universitaria, Ciudad de México, Mexico · Secihti - Centro Interdisciplinario de Ciencias Marinas. IPN., BCS, La Paz, México

Parameter Estimation Stochastic Differential Equations

Mar 25, 2026

Polynomial Speedup in Diffusion Models with the Multilevel Euler-Maruyama Method

We introduce the Multilevel Euler-Maruyama (ML-EM) method compute solutions of SDEs and ODEs using a range of approximators

f^1,\dots,f^k

to the drift

f

with increasing accuracy and computational cost, only requiring a few evaluations of the most accurate

f^k

and many evaluations of the less costly

f^1,\dots,f^{k-1}

. If the drift lies in the so-called Harder than Monte Carlo (HTMC) regime, i.e. it requires

ε^{-γ}

compute to be

ε

-approximated for some

γ>2

, then ML-EM

ε

-approximates the solution of the SDE with

ε^{-γ}

compute, improving over the traditional EM rate of

ε^{-γ-1}

. In other terms it allows us to solve the SDE at the same cost as a single evaluation of the drift. In the context of diffusion models, the different levels

f^{1},\dots,f^{k}

are obtained by training UNets of increasing sizes, and ML-EM allows us to perform sampling with the equivalent of a single evaluation of the largest UNet. Our numerical experiments confirm our theory: we obtain up to fourfold speedups for image generation on the CelebA dataset downscaled to 64x64, where we measure a

γ\approx2.5

. Given that this is a polynomial speedup, we expect even stronger speedups in practical applications which involve orders of magnitude larger networks.

Arthur Jacot

Diffusion Model Inference Acceleration Diffusion Sampling Stochastic Differential Equations Diffusion Models

Mar 24, 2026

Neural ODE and SDE Models for Adaptation and Planning in Model-Based Reinforcement Learning

We investigate neural ordinary and stochastic differential equations (neural ODEs and SDEs) to model stochastic dynamics in fully and partially observed environments within a model-based reinforcement learning (RL) framework. Through a sequence of simulations, we show that neural SDEs more effectively capture the inherent stochasticity of transition dynamics, enabling high-performing policies with improved sample efficiency in challenging scenarios. We leverage neural ODEs and SDEs for efficient policy adaptation to changes in environment dynamics via inverse models, requiring only limited interactions with the new environment. To address partial observability, we introduce a latent SDE model that combines an ODE with a GAN-trained stochastic component in latent space. Policies derived from this model provide a strong baseline, outperforming or matching general model-based and model-free approaches across stochastic continuous-control benchmarks. This work demonstrates the applicability of action-conditional latent SDEs for RL planning in environments with stochastic transitions. Our code is available at: https://github.com/ChaoHan-UoS/NeuralRL

Chao Han, Stefanos Ioannou, Luca Manneschi +4

School of Computer Science, The University of Sheffield, UK · Cancer Research UK National Biomarker Centre, The University of Manchester, UK · School of Chemical, Materials and Biological Engineering, The University of Sheffield, UK +2

Model-Based RL Latent Dynamics Modeling Partially Observable RL Neural ODEs Stochastic Differential Equations Non-Stationary RL

Mar 20, 2026

Goal-oriented learning of stochastic differential equations using error bounds on path-space observables

Stochastic differential equations (SDEs), which serve as the governing equations for dynamical systems in a broad range of applications, can become cost-prohibitive for numerical simulation at scales necessary for quantifying key properties. Surrogate models of the drift function of an SDE, learned from data of the high-fidelity system, are routinely used to increase the efficiency of simulation and prediction of properties. However, standard choices of loss function for learning the surrogate model fail to provide error guarantees in certain path-dependent observables, such as transition times. This paper introduces an error bound for path-space observables and employs it as a novel variational loss for the goal-oriented learning of the drift function of a SDE. We show the error bound holds for a broad class of observables, including mean first hitting times on unbounded time domains. We derive an analytical gradient of the goal-oriented loss by leveraging the formula for Fréchet derivatives of expected path functionals, which remains tractable for implementation in stochastic gradient descent schemes. We demonstrate that surrogate models of overdamped Langevin systems developed via goal-oriented learning achieve improved accuracy in predicting the statistics of a first hitting time observable and robustness to distributional shift in the data.

Joanna Zou, Han Cheng Lie, Youssef Marzouk

Center for Computational Science & Engineering, Massachusetts Institute of Technology, Cambridge, MA · Department of Mathematics, University of Potsdam, Potsdam, Germany

Surrogate Modeling Stochastic Differential Equations

Mar 10, 2026

Symbolic Discovery of Stochastic Differential Equations with Genetic Programming

Automated scientific discovery aims to improve scientific understanding through machine learning. A central approach in this field is symbolic regression, which uses genetic programming or sparse regression to learn interpretable mathematical expressions to explain observed data. Conventionally, the focus of symbolic regression is on identifying ordinary differential equations. The general view is that noise only complicates the recovery of deterministic dynamics. However, explicitly learning a symbolic function of the noise component in stochastic differential equations enhances modelling capacity, increases knowledge gain and enables generative sampling. We introduce a method for symbolic discovery of stochastic differential equations based on genetic programming, jointly optimizing drift and diffusion functions via the maximum likelihood estimate. Our results demonstrate accurate recovery of governing equations, efficient scaling to higher-dimensional systems, robustness to sparsely sampled problems and generalization to stochastic partial differential equations. This work extends symbolic regression toward interpretable discovery of stochastic dynamical systems, contributing to the automation of science in a noisy and dynamic world.

Sigur de Vries, Sander W. Keemink, Marcel A. J. van Gerven

Department of Machine Learning and Neural Computing, Donders Institute for Brain, Cognition and Behaviour, Radboud University, Nijmegen, the Netherlands

Stochastic Differential Equations Symbolic Regression

Mar 10, 2026

Stochastic Port-Hamiltonian Neural Networks: Universal Approximation with Passivity Guarantees

Stochastic port-Hamiltonian systems represent open dynamical systems with dissipation, inputs, and stochastic forcing in an energy based form. We introduce stochastic port-Hamiltonian neural networks, SPH-NNs, which parameterize the Hamiltonian with a feedforward network and enforce skew symmetry of the interconnection matrix and positive semidefiniteness of the dissipation matrix. For Itô dynamics we establish a weak passivity inequality in expectation under an explicit generator condition, stated for a stopped process on a compact set. We also prove a universal approximation result showing that, on any compact set and finite horizon, SPH-NNs approximate the coefficients of a target stochastic port-Hamiltonian system with

C^2

accuracy of the Hamiltonian and yield coupled solutions that remain close in mean square up to the exit time. Experiments on noisy mass spring, Duffing, and Van der Pol oscillators show improved long horizon rollouts and reduced energy error relative to a multilayer perceptron baseline.

Luca Di Persio, Matthias Ehrhardt, Youness Outaleb

Department of Computer Science - College of Mathematics University of Verona, Strada le Grazie 15 - 37134 Verona, Italy · University of Wuppertal, Chair of Applied and Computational Mathematics, Gaußstrasse 20, 42119 Wuppertal, Germany · Doctoral School in Mathematics, University of Trento Via Sommarive, 14 - 38123 Povo, Italy

Port-Hamiltonian Systems Hamiltonian Neural Networks Stochastic Differential Equations Neural Network Approximation Theory Passivity-Based Control

Mar 10, 2026

Differentiable Stochastic Traffic Dynamics: Physics-Informed Generative Modelling in Transportation

Macroscopic traffic flow is stochastic, but the physics-informed deep learning methods currently used in transportation literature embed deterministic PDEs and produce point-valued outputs; the stochasticity of the governing dynamics plays no role in the learned representation. This work develops a framework in which the physics constraint itself is distributional and directly derived from stochastic traffic-flow dynamics. Starting from an Ito-type Lighthill-Whitham-Richards model with Brownian forcing, we derive a one-point forward equation for the marginal traffic density at each spatial location. The spatial coupling induced by the conservation law appears as an explicit conditional drift term, which makes the closure requirement transparent. Based on this formulation, we derive an equivalent deterministic Probability Flow ODE that is pointwise evaluable and differentiable once a closure is specified. Incorporating this as a physics constraint, we then propose a score network with an advection-closure module, trainable by denoising score matching together with a Fokker-Planck residual loss. The resulting model targets a data-conditioned density distribution, from which point estimates, credible intervals, and congestion-risk measures can be computed. The framework provides a basis for distributional traffic-state estimation and for stochastic fundamental-diagram analysis in a physics-informed generative setting.

Wuping Xin

Caliper Corporation

Physics-Informed Generative Modeling Traffic Flow Estimation Physics-Informed Diffusion Models Stochastic Differential Equations

Mar 9, 2026

Sign Identifiability of Causal Effects in Stationary Stochastic Dynamical Systems

We study identifiability in continuous-time linear stationary stochastic differential equations with a known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Therefore, rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrisations compatible with that structure. This leads to a trichotomy of edge-sign identifiability: identifiable, non-identifiable, and partially identifiable. This trichotomy introduces the new notion of partial identifiability to the literature, which we show is a genuine category in our setting. Under a notion of faithfulness, we derive criteria to identify membership of each category for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.

Gijs van Seeventer, Saber Salehkaleybar

1LIACS, Leiden University, Netherlands

Stochastic Differential Equations Structural Causal Models Causal Identifiability

Mar 6, 2026

Random Quadratic Form on a Sphere: Synchronization by Common Noise

We introduce the Random Quadratic Form (RQF): a stochastic differential equation which formally corresponds to the gradient flow of a random quadratic functional on a sphere. While the one-point dynamics of the system is a Brownian motion and thus has no preferred direction, the two-point motion exhibits nontrivial synchronizing behaviour. In this work we study synchronization of the RQF, namely we give both distributional and path-wise characterizations of the solutions by studying invariant measures and random attractors of the system. The RQF model is motivated by the study of the role of linear layers in transformers and illustrates the synchronization by common noise phenomena arising in the simplified models of transformers. In particular, we provide an alternative (independent of self-attention) explanation of the clustering behaviour in deep transformers and show that tokens cluster even in the absence of the self-attention mechanism.

Maximilian Engel, Anna Shalova

Korteweg-de Vries Institute for Mathematics, University of Amsterdam · Department of Mathematics and Computer Science, FU Berlin

Stochastic Differential Equations Dynamical Systems Transformer

Mar 4, 2026

Lyapunov Stability of Stochastic Vector Optimization: Theory and Numerical Implementation

The use of stochastic differential equations in multi-objective optimization has been limited, in practice, by two persistent gaps: incomplete stability analyses and the absence of accessible implementations. We revisit a drift--diffusion model for unconstrained vector optimization in which the drift is induced by a common descent direction and the diffusion term preserves exploratory behavior. The main theoretical contribution is a self-contained Lyapunov analysis establishing global existence, pathwise uniqueness, and non-explosion under a dissipativity condition, together with positive recurrence under an additional coercivity assumption. We also derive an Euler--Maruyama discretization and implement the resulting iteration as a pymoo-compatible algorithm -- pymoo being an open-source Python framework for multi-objective optimization -- with an interactive PymooLab front-end for reproducible experiments. Empirical results on DTLZ2 with objective counts from three to fifteen indicate a consistent trade-off: compared with established evolutionary baselines, the method is less competitive in low-dimensional regimes but remains a viable option under restricted evaluation budgets in higher-dimensional settings. Taken together, these observations suggest that stochastic drift--diffusion search occupies a mathematically tractable niche alongside population-based heuristics -- not as a replacement, but as an alternative whose favorable properties are amenable to rigorous analysis.

Thiago Santos, Sebastiao Xavier

Department of Mathematics, Federal University of Ouro Preto, UFOP, Ouro Preto, Brazil

Multi-Objective Optimization Stochastic Differential Equations Dynamical Systems

Mar 4, 2026

Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme

Low-dimensional structure in real-world data plays an important role in the success of generative models, which motivates diffusion models defined on intrinsic data manifolds. Such models are driven by stochastic differential equations (SDEs) on manifolds, which raises the need for convergence theory of numerical schemes for manifold-valued SDEs. In Euclidean space, the Euler--Maruyama (EM) scheme achieves strong convergence with order

1/2

, but an analogous result for manifold discretizations is less understood in general settings. In this work, we study a geometric version of the EM scheme for SDEs on Riemannian manifolds and prove strong convergence with order

1/2

under geometric and regularity conditions. As an application, we obtain a Wasserstein bound for sampling on manifolds via the geometric EM discretization of Riemannian Langevin dynamics.

Zhiyuan Zhan, Masashi Sugiyama

The University of Tokyo, Japan · RIKEN Center for AIP, Japan

Stochastic Differential Equations Langevin Dynamics

Feb 27, 2026

Neural Diffusion Intensity Models for Point Process Data

Cox processes model overdispersed point process data via a latent stochastic intensity, but both nonparametric estimation of the intensity model and posterior inference over intensity paths are typically intractable, relying on expensive MCMC methods. We introduce Neural Diffusion Intensity Models, a variational framework for Cox processes driven by neural SDEs. Our key theoretical result, based on enlargement of filtrations, shows that conditioning on point process observations preserves the diffusion structure of the latent intensity with an explicit drift correction. This guarantees the variational family contains the true posterior, so that ELBO maximization coincides with maximum likelihood estimation under sufficient model capacity. We design an amortized encoder architecture that maps variable-length event sequences to posterior intensity paths by simulating the drift-corrected SDE, replacing repeated MCMC runs with a single forward pass. Experiments on synthetic and real-world data demonstrate accurate recovery of latent intensity dynamics and posterior paths, with orders-of-magnitude speedups over MCMC-based methods.

Xinlong Du, Harsha Honnappa, Vinayak Rao

Edwardson School of Industrial Engineering, Purdue University · Department of Statistics, Purdue University

Temporal Point Processes Variational Inference Stochastic Differential Equations

Feb 23, 2026

Generative Modeling via Kernelized Stochastic Interpolants

We develop a kernel method for generative modeling within the stochastic interpolant framework, replacing neural network training with linear systems. The drift of the generative SDE is

\hat b_t(x) = \nablaφ(x)^\topη_t

, where

η_t \in \mathbb{R}^P

solves a

P\times P

system computable from data, with

P

independent of the data dimension

d

. Since estimates are inexact, the diffusion coefficient

D_t

affects sample quality; the optimal

D_t^*

from Girsanov diverges at

t=0

, but this poses no difficulty and we develop an integrator that handles it seamlessly. The framework accommodates diverse feature maps: scattering transforms, pretrained generative models, etc, enabling generation and model combination without neural network training. We demonstrate the approach on financial time series, turbulence, and image generation.

Florentin Coeurdoux, Etienne Lempereur, Nathanaël Cuvelle-Magar +2

Capital Fund Management, Paris, France · Département d’informatique, ENS, Université PSL, Paris, France · Collège de France, Paris, France +2

Stochastic Differential Equations Stochastic Interpolants Generative Modeling Kernel Methods

Feb 19, 2026

Drift Estimation for Stochastic Differential Equations with Denoising Diffusion Models

We study the estimation of time-homogeneous drift functions in multivariate stochastic differential equations with known diffusion coefficient, from multiple trajectories observed at high frequency over a fixed time horizon. We formulate drift estimation as a denoising problem conditional on previous observations, and propose an estimator of the drift function which is a by-product of training a conditional diffusion model capable of simulating new trajectories dynamically. Across different drift classes, the proposed estimator was found to match classical methods in low dimensions and remained consistently competitive in higher dimensions, with gains that cannot be attributed to architectural design choices alone.

Marcos Tapia Costa, Nikolas Kantas, George Deligiannidis

Department of Mathematics, Imperial College London, UK · Department of Statistics, University of Oxford, UK

Stochastic Differential Equations

Feb 17, 2026

Steering Dynamical Regimes of Diffusion Models by Breaking Detailed Balance

We show that deliberately breaking detailed balance in generative diffusion processes can accelerate the reverse process without changing the stationary distribution. Considering the Ornstein--Uhlenbeck process, we decompose the dynamics into a symmetric component and a non-reversible anti-symmetric component that generates rotational probability currents. We then construct an exponentially optimal non-reversible perturbation that improves the long-time relaxation rate while preserving the stationary target. We analyze how such non-reversible control reshapes the macroscopic dynamical regimes of the phase transitions recently identified in generative diffusion models. We derive a general criterion for the speciation time and show that suitable non-reversible perturbations can accelerate speciation. In contrast, the collapse transition is governed by a trace-controlled phase-space contraction mechanism that is fixed by the symmetric component, and the corresponding collapse time remains unchanged under anti-symmetric perturbations. Numerical experiments on Gaussian mixture models support these findings.

Haiqi Lu, Ying Tang

Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China · Yingcai Honor College, University of Electronic Science and Technology of China, Chengdu 611731, China · School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China +2

Diffusion Models Diffusion Sampling Diffusion Model Acceleration Stochastic Differential Equations

Feb 16, 2026

Drift-Diffusion Matching: Embedding dynamics in latent manifolds of asymmetric neural networks

Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits, yet classical results, such as Hopfield's model of associative memory, rely on symmetric connectivity that restricts network dynamics to gradient-like flows. In contrast, biological networks support rich time-dependent behaviour facilitated by their asymmetry. Here we introduce a general framework, which we term drift-diffusion matching, for training continuous-time RNNs to represent arbitrary, nonlinear stochastic differential equations (SDEs), with given drift and diffusion coefficients, within a low-dimensional latent subspace. Allowing asymmetric connectivity, we show that RNNs can faithfully embed the drift and diffusion of a given SDE, including nonlinear and nonequilibrium dynamics such as chaotic attractors. As an application, we construct RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, we introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. Our results extend attractor neural network theory beyond equilibrium, showing that asymmetric neural populations can implement a broad class of dynamical computations within low-dimensional manifolds, unifying ideas from associative memory, nonequilibrium statistical mechanics, and neural computation.

Ramón Nartallo-Kaluarachchi, Renaud Lambiotte, Alain Goriely

Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, United Kingdom · Centre for Eudaimonia and Human Flourishing, University of Oxford, 7 Stoke Pl, Oxford, OX3 9BX, United Kingdom · Complexity Science Hub, Metternichgasse 8, Vienna, 1030, Austria

Recurrent Neural Networks Dynamical Systems Latent Dynamics Modeling Stochastic Differential Equations Associative Memory