Mitigating Propagation Failures in PINNs using Evolutionary Sampling

Abstract

Despite the success of physics-informed neural networks (PINNs) in approximating partial differential equations (PDEs), it is known that PINNs can sometimes fail to converge to the correct solution in problems involving complicated PDEs. This is reflected in several recent studies on characterizing and mitigating the ``failure modes'' of PINNs. While most of these studies have focused on balancing loss functions or adaptively tuning PDE coefficients, what is missing is a thorough understanding of the connection between failure modes of PINNs and sampling strategies used for training PINNs. In this paper, we provide a novel perspective of failure modes of PINNs by hypothesizing that the training of PINNs rely 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. We additionally demonstrate that propagation failures are characterized by highly imbalanced PDE residual fields where very high residuals are observed over very narrow regions. To mitigate propagation failures, we propose a novel evolutionary sampling (Evo) method that can incrementally accumulate collocation points in regions of high PDE residuals with little to no computational overhead. We provide an extension of Evo to respect the principle of causality while solving time-dependent PDEs. We theoretically analyze the behavior of Evo and empirically demonstrate its efficacy and efficiency in comparison with baselines on a variety of PDE problems.Comment: 34 pages, 46 figures, 2 table