Self-Supervised Learning with Lie Symmetries for Partial Differential
  Equations
Authors: name: Grégoire Mialon; name: Quentin Garrido; name: Hannah Lawrence; name: Danyal Rehman; name: Yann LeCun; name: Bobak T. Kiani
Published: 2023-07-11 16:52:22

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.