Variational Physics Informed Neural Networks: the role of quadratures and test functions
This work provides incremental insights into optimizing VPINNs for computational physics, specifically targeting researchers in scientific machine learning.
The paper analyzes how quadrature precision and test function polynomial degree affect the convergence rate of Variational Physics Informed Neural Networks (VPINNs) for solving elliptic boundary-value problems, finding that using low-degree test functions with high-precision quadratures yields the highest error decay rates for smooth solutions.
In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.