Abstract: Artificial neural networks and modern accelerators provide a platform for developing mesh-free approaches to solving partial differential equations, particularly in high-dimensional settings where classical grid-based methods become infeasible. These methods typically reduce PDE solving to a nonlinear optimization problem: adjust the parameters of a network function to minimize the residual of the differential operator. This residual minimization viewpoint is the subject of the talk.

 

We begin by establishing necessary conditions under which residual minimization can recover the solution of a well-posed initial-boundary value problem. Through elementary examples, we examine what goes wrong when these conditions fail and why the resulting optimization landscapes give rise to loss-function fallacies — situations where small training loss does not imply small solution error.

 

In the second half, which will be given at Fu Jen Catholic University, we show how classical numerical principles both diagnose these pitfalls and suggest superior alternatives, through two applications: (1) Hamilton-Jacobi equations, where neural solvers struggle with viscosity solutions in high dimensions, and (2) integral equations, where the dense, singular matrices arising from quadrature demand structure-aware inversion. The talk aims to present  a framework for deciding when to trust neural solvers and how to use classical theory to build more robust computational tools.

 

Organizers: Yu-Chen Shu (NCKU), Ming-Cheng Shiue (NYCU), Chien-Chang Yen (FJCU).