mlsanityData to 14 Oct 2026

Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics

arXiv 2110.10249 ↗First posted 19 Oct 2021cs.LGcited by 47

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Abstract (arXiv)

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural architecture to learn solution operators of PDEs with (possibly stochastic) forcing from partially observed data. The proposed Neural SPDE model provides an extension to two popular classes of physics-inspired architectures. On the one hand, it extends Neural CDEs and variants -- continuous-time analogues of RNNs -- in that it is capable of processing incoming sequential information arriving at arbitrary spatial resolutions. On the other hand, it extends Neural Operators -- generalizations of neural networks to model mappings between spaces of functions -- in that it can parameterize solution operators of SPDEs depending simultaneously on the initial condition and a realization of the driving noise. By performing operations in the spectral domain, we show how a Neural SPDE can be evaluated in two ways, either by calling an ODE solver (emulating a spectral Galerkin scheme), or by solving a fixed point problem. Experiments on various semilinear SPDEs, including the stochastic Navier-Stokes equations, demonstrate how the Neural SPDE model is capable of learning complex spatiotemporal dynamics in a resolution-invariant way, with better accuracy and lighter training data requirements compared to alternative models, and up to 3 orders of magnitude faster than traditional solvers.

Key facts. Cited by 47 papers in the map (data through 14 Oct 2026). Builds on 16 of its 52 references that are in the map. First posted on arXiv 19 Oct 2021.

Builds on

16 of 52 references.

References · 16

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  5. 2019Deep Equilibrium Modelscited by 615
  6. 2020Neural Controlled Differential Equations for Irregular Time Seriescited by 506
  7. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  8. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  9. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
  10. 2021Neural SDEs as Infinite-Dimensional GANscited by 161
  11. 2020Neural Rough Differential Equations for Long Time Seriescited by 143
  12. 2019Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noisecited by 101
4 more
  1. 2021Efficient and Accurate Gradients for Neural SDEscited by 71
  2. 2020The Signature Kernel is the solution of a Goursat PDEcited by 58
  3. 2021Policy Analysis using Synthetic Controls in Continuous-Timecited by 18
  4. 2021Feature Engineering with Regularity Structurescited by 7

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Citing papers per year: 2022: 1; 2023: 15; 2024: 12; 2025: 8; 2026: 11.

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  1. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  2. 2024Theoretical Foundations of Deep Selective State-Space Modelscited by 74
  3. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 45
  4. 2023Non-adversarial training of Neural SDEs with signature kernel scorescited by 37
  5. 2023SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biasescited by 27
  6. 2023Neural signature kernels as infinite-width-depth-limits of controlled ResNetscited by 25
  7. 2023Global universal approximation of functional input maps on weighted spacescited by 19
  8. 2023Solving Inverse Physics Problems with Score Matchingcited by 16
  9. 2023A Brief Survey on the Approximation Theory for Sequence Modellingcited by 15
  10. 2024Parametric model reduction of mean-field and stochastic systems via higher-order action matchingcited by 9
  11. 2023Infinite-dimensional reservoir computingcited by 7
  12. 2022Towards Size-Independent Generalization Bounds for Deep Operator Netscited by 6
35 more
  1. 2023Equivariant Neural Simulators for Stochastic Spatiotemporal Dynamicscited by 6
  2. 2024Solving stochastic partial differential equations using neural networks in the Wiener chaos expansioncited by 6
  3. 2025pySigLib -- Fast Signature-Based Computations on CPU and GPUcited by 6
  4. 2024Learning with SASQuaTCh: a Novel Variational Quantum Transformer Architecture with Kernel-Based Self-Attentioncited by 5
  5. 2025FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulationcited by 5
  6. 2026Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equationscited by 5
  7. 2024Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilationcited by 4
  8. 2025Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamicscited by 4
  9. 2023Learning Space-Time Continuous Neural PDEs from Partially Observed Statescited by 3
  10. 2024Neural Quantum Propagators for Driven-Dissipative Quantum Dynamicscited by 3
  11. 2023Finding Short Signals in Long Irregular Time Series with Continuous-Time Attention Policy Networkscited by 2
  12. 2023Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUpcited by 2
  13. 2024Learning Spatiotemporal Dynamical Systems from Point Process Observationscited by 2
  14. 2025Noisy PDE Training Requires Bigger PINNscited by 2
  15. 2025Deep Neural Networks Inspired by Differential Equationscited by 2
  16. 2026TRIE: An Evaluation Framework for Stochastic PDE Surrogatescited by 2
  17. 2026StocBench: A Benchmark for Generative Modeling of Stochastic Dynamicscited by 2
  18. 2024Neural stochastic Volterra equations: learning path-dependent dynamicscited by 1
  19. 2025Chaos into Order: Neural Framework for Expected Value Estimation of Stochastic Partial Differential Equationscited by 1
  20. 2025SPDEBench: An Extensive Benchmark for Learning Stochastic PDEscited by 1
  21. 2026Latent-Variable Learning of SPDEs via Wiener Chaoscited by 1
  22. 2026Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equationscited by 1
  23. 2023Continuous Time Evidential Distributions for Irregular Time Seriescited by 0
  24. 2023On the Parallel Complexity of Multilevel Monte Carlo in Stochastic Gradient Descentcited by 0
  25. 2024Log-PDE Methods for Rough Signature Kernelscited by 0
  26. 2024Neural Controlled Differential Equations with Quantum Hidden Evolutionscited by 0
  27. 2024Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditionscited by 0
  28. 2024Deep Optimal Sensor Placement for Black Box Stochastic Simulationscited by 0
  29. 2025Novelty detection on path spacecited by 0
  30. 2026Generative Neural Operators through Diffusion Last Layercited by 0
  31. 2026Uncertainty-Calibrated Spatiotemporal Field Diffusion with Sparse Supervisioncited by 0
  32. 2026Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equationscited by 0
  33. 2026Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorizationcited by 0
  34. 2026Improving Function Space Flow Matching with Kernel Optimal Transportcited by 0
  35. 2026Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equationscited by 0