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Despite the success of physics-informed neural networks (PINNs) inapproximating partial differential equations (PDEs), PINNs can sometimes failto converge to the correct solution in problems involving complicated PDEs.This is reflected in several recent studies on characterizing the "failuremodes" of PINNs, although a thorough understanding of the connection betweenPINN failure modes and sampling strategies is missing. In this paper, weprovide a novel perspective of failure modes of PINNs by hypothesizing thattraining PINNs relies on successful "propagation" of solution from initialand/or boundary condition points to interior points. We show that PINNs withpoor sampling strategies can get stuck at trivial solutions if there arepropagation failures, characterized by highly imbalanced PDE residual fields.To mitigate propagation failures, we propose a novel Retain-Resample-Releasesampling (R3) algorithm that can incrementally accumulate collocation points inregions of high PDE residuals with little to no computational overhead. Weprovide an extension of R3 sampling to respect the principle of causality whilesolving time-dependent PDEs. We theoretically analyze the behavior of R3sampling and empirically demonstrate its efficacy and efficiency in comparisonwith baselines on a variety of PDE problems.

Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling