Is L 2 Physics Informed Loss Always Suitable for Training Physics Informed Neural Network?

The Physics-Informed Neural Network (PINN) approach is a new and promising way to solve partial differential equations using deep learning. The L 2 Physics-Informed Loss is the de-facto standard in training Physics-Informed Neural Networks. In this paper, we challenge this common practice by investigating the relationship between the loss function and the approximation quality of the learned solution. In particular, we leverage the concept of stability in the literature of partial differential equation to study the asymptotic behavior of the learned solution as the loss approaches zero. With this concept, we study an important class of high-dimensional non-linear PDEs in optimal control, the Hamilton-Jacobi-Bellman (HJB) Equation, and prove that for general L p Physics-Informed Loss, a wide class of HJB equation is stable only if p is sufficiently large.