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

arXiv: https://arxiv.org/abs/2110.10249 · HTML: https://mlsanity.com/paper/2110.10249/

## 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 (most cited first, 16 of 16)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [Policy Analysis using Synthetic Controls in Continuous-Time](https://mlsanity.com/paper/2102.01577/) (2021)
- [Feature Engineering with Regularity Structures](https://mlsanity.com/paper/2108.05879/) (2021)

## Built on by (most cited first, 30 of 47)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases](https://mlsanity.com/paper/2308.13212/) (2023)
- [Neural signature kernels as infinite-width-depth-limits of controlled ResNets](https://mlsanity.com/paper/2303.17671/) (2023)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Solving Inverse Physics Problems with Score Matching](https://mlsanity.com/paper/2301.10250/) (2023)
- [A Brief Survey on the Approximation Theory for Sequence Modelling](https://mlsanity.com/paper/2302.13752/) (2023)
- [Parametric model reduction of mean-field and stochastic systems via higher-order action matching](https://mlsanity.com/paper/2410.12000/) (2024)
- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [Towards Size-Independent Generalization Bounds for Deep Operator Nets](https://mlsanity.com/paper/2205.11359/) (2022)
- [Equivariant Neural Simulators for Stochastic Spatiotemporal Dynamics](https://mlsanity.com/paper/2305.14286/) (2023)
- [Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion](https://mlsanity.com/paper/2411.03384/) (2024)
- [pySigLib -- Fast Signature-Based Computations on CPU and GPU](https://mlsanity.com/paper/2509.10613/) (2025)
- [Learning with SASQuaTCh: a Novel Variational Quantum Transformer Architecture with Kernel-Based Self-Attention](https://mlsanity.com/paper/2403.14753/) (2024)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations](https://mlsanity.com/paper/2601.01021/) (2026)
- [Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilation](https://mlsanity.com/paper/2402.15656/) (2024)
- [Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics](https://mlsanity.com/paper/2506.18339/) (2025)
- [Learning Space-Time Continuous Neural PDEs from Partially Observed States](https://mlsanity.com/paper/2307.04110/) (2023)
- [Neural Quantum Propagators for Driven-Dissipative Quantum Dynamics](https://mlsanity.com/paper/2410.16091/) (2024)
- [Finding Short Signals in Long Irregular Time Series with Continuous-Time Attention Policy Networks](https://mlsanity.com/paper/2302.04052/) (2023)
- [Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp](https://mlsanity.com/paper/2310.05169/) (2023)
- [Learning Spatiotemporal Dynamical Systems from Point Process Observations](https://mlsanity.com/paper/2406.00368/) (2024)
- [Noisy PDE Training Requires Bigger PINNs](https://mlsanity.com/paper/2507.06967/) (2025)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [TRIE: An Evaluation Framework for Stochastic PDE Surrogates](https://mlsanity.com/paper/2607.00196/) (2026)
- [StocBench: A Benchmark for Generative Modeling of Stochastic Dynamics](https://mlsanity.com/paper/2608.22309/) (2026)
- [Neural stochastic Volterra equations: learning path-dependent dynamics](https://mlsanity.com/paper/2407.19557/) (2024)

## 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.