Lie Point Symmetry Data Augmentation for Neural PDE Solvers

2022/02/15 by J. Brandstetter, Johannes Brandstetter, Max Welling +4 · 17 citations
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Artificial intelligence #Artificial neural network #Computer science #Context (archaeology) #Geometry #Mathematical analysis #Mathematics #Model Reduction and Neural Networks #Partial differential equation #Point (geometry) #Solver #Symmetry (geometry) #cs.CV #cs.LG

paper · pdf · doi:10.48550/arxiv.2202.07643

published in arXiv (Cornell University) (Cornell University) · Published at ICML 2022, Github: https://github.com/brandstetter-johannes/LPSDA

openalex publication_date 2022/02/15 · arxiv created 2022/05/29 · arxiv updated 2022/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

Neural networks are increasingly being used to solve partial differential equations (PDEs), replacing slower numerical solvers. However, a critical issue is that neural PDE solvers require high-quality ground truth data, which usually must come from the very solvers they are designed to replace. Thus, we are presented with a proverbial chicken-and-egg problem. In this paper, we present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity -- Lie point symmetry data augmentation (LPSDA). In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations, based on the Lie point symmetry group of the PDEs in question, something not possible in other application areas. We present this framework and demonstrate how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.

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• Scale-Consistent Learning for Partial Differential Equations
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• Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning
• Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates
• Governing Equation Discovery from Data Based on Differential Invariants
• LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems
• Beyond Accuracy: EcoL2 Metric for Sustainable Neural PDE Solvers
• Discovering Symbolic Differential Equations with Symmetry Invariants
• Transforming physics-informed machine learning to convex optimization

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