# SPDEBench: An Extensive Benchmark for Learning Stochastic PDEs

arXiv: https://arxiv.org/abs/2505.18511 · HTML: https://mlsanity.com/paper/2505.18511/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 46 references that are in the map.
- First posted on arXiv 24 May 2025.

## Builds on (most cited first, 16 of 16)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Sig-Wasserstein GANs for Time Series Generation](https://mlsanity.com/paper/2111.01207/) (2021)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics](https://mlsanity.com/paper/2110.10249/) (2021)
- [Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs](https://mlsanity.com/paper/2204.06255/) (2022)
- [Feature Engineering with Regularity Structures](https://mlsanity.com/paper/2108.05879/) (2021)
- [Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion](https://mlsanity.com/paper/2411.03384/) (2024)

## Built on by (most cited first, 1 of 1)

- [Tangent Schr"odinger Bridge Matching: Learning Stochastic Transport with Mechanistic Sensitivities](https://mlsanity.com/paper/2610.02906/) (2026)

## Abstract (arXiv)

Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modeling physical processes with rough spatio-temporal dynamics, such as turbulence flows, superconductors, and quantum dynamics. Although machine learning (ML)-based surrogate models have shown promise for efficiently approximating such dynamics, progress remains limited by the lack of a unified benchmark with controlled data generation and comprehensive evaluation. This gap is particularly significant for singular SPDEs, for which benchmark datasets are largely unavailable and reliable simulation requires numerically delicate schemes based on renormalization. Moreover, subtle differences in data-generation procedures, such as noise approximation, basis choice, and the inclusion of renormalization, can significantly affect the resulting datasets and, consequently, model evaluation. We introduce SPDEBench, the first unified benchmark for ML-based SPDE learning. SPDEBench provides ready-to-use datasets for physically and mathematically significant SPDEs on 1-3D domains with periodic or Dirichlet boundary condition. Both regular and singular SPDEs are taken into consideration. SPDEBench also incorporates representative ML baselines in operator learning, together with 7 evaluation metrics, including Sobolev and distributional metrics beyond the standard L2-error. Supported by SPDEBench, we conduct systematic evaluations of model accuracy, robustness, and out-of-distribution generalization under controlled data variations. Our numerical results show that SPDE-aware architectures generally achieve stronger performance than generic operator-learning baselines. These findings establish SPDEBench as a reproducible and extensible resource, paving pathway for principled benchmarking and architecture design for stochastic spatio-temporal dynamics.