Self-Supervised Learning with Lie Symmetries for Partial Differential Equations

Published: 21 Sept 2023, Last Modified: 02 Nov 2023NeurIPS 2023 posterEveryoneRevisionsBibTeX
Keywords: Self-supervised learning, partial differential equations, Lie symmetries, data augmentation
TL;DR: We apply self-supervised learning to partial differential equations, using the equations' Lie point symmetries as augmentations.
Abstract: Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated training data tailored to a given setting, one may instead wish to learn useful information from heterogeneous sources, or from real dynamical systems observations that are messy or incomplete. In this work, we learn general-purpose representations of PDEs from heterogeneous data by implementing joint embedding methods for self-supervised learning (SSL), a framework for unsupervised representation learning that has had notable success in computer vision. Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs.
Supplementary Material: pdf
Submission Number: 4783