Challenges in automatic differentiation and numerical integration in physics-informed neural networks modelling

In this paper, we numerically examine the precision challenges that emerge in automatic differentiation and numerical integration in various tasks now tackled by physics-informed neural networks (PINNs). Specifically, we illustrate how ill-posed problems or inaccurately computed functions can cause serious precision issues in differentiation and integration. A major difficulty lies in detecting these problems. A simple large-scale view of the function or good-looking loss functions or convergence results may not reveal any potential errors, and the resulting outcomes are often mistakenly considered correct. To address this, it is critical to determine whether standard double-precision arithmetic suffices or if higher precision is necessary. Three problematic use-cases for solving differential equations using PINNs are analysed in detail. For the case requiring numerical integration, we also evaluate several numerical quadrature methods and suggest particular numerical analysis steps to choose the most suitable method.