We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to other PDEs. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the two- and three-dimensional settings.

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. However, a critical question remains: How do we validate that PINNs accurately solve these PDEs? This talk explores the types of mathematical validation required to ensure that PINNs can reliably approximate solutions to PDEs. We will discuss the conditions under which PINNs can converge to the correct solution, the relationship between minimizing residuals and achieving accurate results, and the role of optimization algorithms in this process. Our goal is to provide a clear understanding of the theoretical foundations needed to trust PINNs in practical applications while addressing the challenges in this emerging field.