Adaptive collocation and residual verification for engineering physics-informed neural networks

Abstract

Adaptive collocation is often described as placing training points where a physics-informed neural network needs them most. That description hides two distinct operations: estimating a fixed residual objective more efficiently, and deliberately changing the objective to emphasize selected regions. This review derives the distinction using importance sampling, examines the difference between residual magnitude and gradient variance, and develops an analytical hot-spot example to quantify missed-region risk. Applications to wake dynamics, thermal layers, solid mechanics, and interfaces show why sampling must be tied to geometry and engineering quantities of interest. Independent verification, support-preserving sampling, causal time allocation, and complete computational accounting are assembled into a proposed evaluation protocol. The worked probability model connects sampling choices to the risk of missing a critical region.