Neural stochastic PDEs: resolution-invariant learning of continuous spatiotemporal dynamics
File(s)
Accepted version
Author(s)
Salvi, Cristopher
Lemercier, Maud
Gerasimovics, Andris
Type
Conference Paper
Abstract
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.
Date Issued
2022-11-28
Date Acceptance
2022-09-14
Citation
Advances in Neural Information Processing Systems 36 (NeurIPS 2022), 2022, pp.1-12
Publisher
NeurIPS
Start Page
1
End Page
12
Journal / Book Title
Advances in Neural Information Processing Systems 36 (NeurIPS 2022)
Copyright Statement
© 2022 The Author(s).
Identifier
https://proceedings.neurips.cc/paper_files/paper/2022/hash/091166620a04a289c555f411d8899049-Abstract-Conference.html
Source
Thirty-sixth Conference on Neural Information Processing Systems (NeurIPS 2022)
Publication Status
Published
Start Date
2022-11-28
Finish Date
2022-12-09
Coverage Spatial
New Orleans, USA
Date Publish Online
2022-11-28