Fully Discrete Analysis of the Galerkin POD Neural Network Approximation with Application to 3D Acoustic Wave Scattering

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. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are publicly available on the bonndata fileservers at https://doi.org/10.60507/FK2/OJUHWY [ 22 ]. [Formula: see text]