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.

Solving PDEs by variational physics-informed neural networks: an a posteriori error analysis

“…However, the forthcoming formulation of the discretized problem and the a posteriori error analysis can be extended to cover the case of mixed Dirichlet-Neumann boundary conditions, namely u = g on D , μ∂ n u = ψ on N , with D ∪ N = . We refer to [13,15] for the general case.…”

“…Uniqueness may not occur. Indeed, any solution u NN of ( 8) annihilates the loss function, hence it is a solution of ( 11); such a solution may not be unique, since the set of equations ( 8) may be underdetermined (in particular, for f = 0 one may obtain a non-zero u NN , see [13,Sect. 6.3]).…”

“…Remark 2 (Discretization with interpolation) In order to reduce and control the randomic effects related to the use of a network depending upon a large number of weights, in [13] we proposed to locally project the neural network upon a space of polynomials, before computing the loss function.…”

“…In [13] we derived an a priori error estimate for the error u − ũNN H V , and we documented the error decay as h → 0, which turns out to have a more regular behavior that the error u − u NN V , although the latter is usually smaller. The subsequent a posteriori error analysis could be extended to give a control on the error produced by ũNN H as well.…”

“…To this respect, we considered in [13] a Petrov-Galerkin framework in which trial functions are defined by means of neural networks, whereas test functions are made of continuous, piecewise linear functions on a triangulation of the domain. Relying on an inf-sup condition between spaces of piecewise polynomial functions, we derived an a priori error estimate in the energy norm between the exact solution of an elliptic boundary-value problem and a high-order interpolant of a deep neural network, which minimizes the loss function.…”