Variational Physics Informed Neural Networks: the role of quadratures and test functions

Abstract

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.

20 pages, 22 figures

References in corpus (7)

• Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
• Variational Physics Informed Neural Networks: the role of quadratures and test functions
• Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imaging
• VarNet: Variational Neural Networks for the Solution of Partial Differential Equations
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• Meshfree Variational Physics Informed Neural Networks (MF-VPINN): an adaptive training strategy
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