Self-Supervised Learning with Lie Symmetries for Partial Differential Equations

TL;DR

This paper introduces a self-supervised learning framework leveraging Lie symmetries to learn general-purpose representations of PDEs from heterogeneous data, enhancing invariant task performance and neural PDE solver efficiency.

Contribution

It proposes a novel SSL method incorporating Lie symmetries for PDEs, enabling learning from diverse data sources and improving downstream task performance.

Findings

Outperforms baseline methods in invariant PDE tasks

Enhances neural solver time-stepping performance

Facilitates learning from real, messy data

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. Code: https://github.com/facebookresearch/SSLForPDEs.