Message Passing Neural PDE Solvers

By Johannes Brandstetter, Daniel E. Worrall, Max Welling. International Conference on Learning Representations, 2022

458 citations recorded by the source corpus.

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.

Subfields: Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/be8d39424a9010bfc0805385cc91edee383c2e24

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In Statistical and Nonlinear Physics

Citations in the corpus

PDEBENCH: An Extensive Benchmark for Scientific Machine Learning (2022, methodology): “parameters, varying initial conditions, and proper treatment of complex boundary conditions are still open challenges in Scientific ML [21, 4, 2, 29].”
Transformer for Partial Differential Equations' Operator Learning (2022, methodology): “…and straightforward approach is to follow an Encoder-Process-Decoder (EPD) scheme to map the input solution at time t to the solution at next time step (Brandstetter et al., 2022; Stachenfeld et al., 2022; Pfaﬀ et al., 2020; Sanchez-Gonzalez et al., 2020; Pant et al., 2021; Hsieh et al., 2019).”
A foundation model for the Earth system (2025, unknown): “To save memory, we use the ‘pushforward trick’ 54 , which propagates gradients only through the last roll-out step.”
Scientific discovery in the age of artificial intelligence (2023, methodology): “This can be achieved by combining neural differential equations with graph neural networks to discretize arbitrary by graph partitioning 182 .”