  Physics-informed neural networks (PINNs) have emerged as an effective
technique for solving PDEs in a wide range of domains. It is noticed, however,
the performance of PINNs can vary dramatically with different sampling
procedures. For instance, a fixed set of (prior chosen) training points may
fail to capture the effective solution region (especially for problems with
singularities). To overcome this issue, we present in this work an adaptive
strategy, termed the failure-informed PINNs (FI-PINNs), which is inspired by
the viewpoint of reliability analysis. The key idea is to define an effective
failure probability based on the residual, and then, with the aim of placing
more samples in the failure region, the FI-PINNs employs a failure-informed
enrichment technique to adaptively add new collocation points to the training
set, such that the numerical accuracy is dramatically improved. In short,
similar as adaptive finite element methods, the proposed FI-PINNs adopts the
failure probability as the posterior error indicator to generate new training
points. We prove rigorous error bounds of FI-PINNs and illustrate its
performance through several problems.
