Fully discrete analysis of the Galerkin POD neural network approximation with application to 3D acoustic wave scattering

TL;DR

This paper presents a fully discrete error analysis of the Galerkin POD neural network method for approximating parametric maps, including discretization errors, and demonstrates its application to 3D acoustic wave scattering.

Contribution

It provides the first fully discrete error bounds for the Galerkin POD-NN approach, accounting for discretization and sampling errors, with practical bounds for implementation.

Findings

Error bounds depend on POD tolerance, ranks, and NN parameters.

Method achieves convergence rates comparable to quasi Monte Carlo sampling.

Applied successfully to a 3D acoustic scattering problem.

Abstract

In this work, we consider the approximation of parametric maps using the so-called Galerkin POD-NN method. This technique combines the computation of a reduced basis via proper orthogonal decomposition (POD) and artificial neural networks (NNs) for the construction of fast surrogates of said parametric maps. In contrast to the existing literature, which has studied the approximation properties of this kind of architecture on a continuous level, we provide a fully discrete error analysis of this approach. More precisely, our estimates also account for discretization errors during the construction of the NN architecture. We consider the number of reduced basis in the approximation of the solution manifold, truncation in the parameter space, and, most importantly, the number of samples in the computation of the reduced space, together with the effect of the use of NNs in the approximation of the reduced coefficients. Following this error analysis, we provide a-priori bounds on the required POD tolerance, the resulting POD ranks, and NN parameters to maintain the order of convergence of quasi Monte Carlo sampling techniques. We conclude this work by showcasing the applicability of this method through a practical industrial application: the sound-soft acoustic scattering problem by a parametrically defined scatterer in three physical dimensions.