Abstract
Despite the success of physics-informed neural networks (PINNs) in approximating partial differential equations (PDEs), PINNs can sometimes fail to converge to the correct solution in problems involving complicated PDEs. This is reflected in several recent studies on characterizing the “failure modes” of PINNs, although a thorough understanding of the connection between PINN failure modes and sampling strategies is missing. In this paper, we provide a novel perspective of failure modes of PINNs by hypothesizing that training PINNs relies on successful “propagation” of solution from initial and/or boundary condition points to interior points. We show that PINNs with poor sampling strategies can get stuck at trivial solutions if there are propagation failures, characterized by highly imbalanced PDE residual fields. To mitigate propagation failures, we propose a novel Retain-Resample-Release sampling (R3) algorithm that can incrementally accumulate collocation points in regions of high PDE residuals with little to no computational overhead. We provide an extension of R3 sampling to respect the principle of causality while solving time-dependent PDEs. We theoretically analyze the behavior of R3 sampling and empirically demonstrate its efficacy and efficiency in comparison with baselines on a variety of PDE problems.
Original language | English |
---|---|
Pages (from-to) | 7264-7302 |
Number of pages | 39 |
Journal | Proceedings of Machine Learning Research |
Volume | 202 |
State | Published - 2023 |
Event | 40th International Conference on Machine Learning, ICML 2023 - Honolulu, United States Duration: Jul 23 2023 → Jul 29 2023 |
Funding
We gratefully acknowledge the support of the National Science Foundation (NSF) under Grant #2213550 and #2107332, which provided funding for this research. We would also like to acknowledge the support of Virginia Tech Advanced Research Computing (ARC) for providing us with necessary resources for our study.