Self-Supervised Learning with Lie Symmetries for Partial Differential Equations
This work addresses the need for efficient PDE solvers in science and engineering by enabling learning from messy or incomplete data, though it is incremental as it adapts existing SSL methods to a new domain.
The paper tackles the problem of learning general-purpose representations of partial differential equations (PDEs) from heterogeneous data using self-supervised learning, resulting in improved performance over baselines in tasks like coefficient regression and neural solver time-stepping.
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.