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

TL;DR

The paper develops a fully discrete error analysis for the Galerkin POD-NN method to approximate parametric maps arising from PDEs and boundary integral equations, explicitly accounting for discretization in both the reduced basis and the neural-network surrogate. By leveraging $(oldsymbol{b},p,oldsymbol{ u})$-holomorphy and quasi-Monte Carlo sampling, it derives a-priori bounds that balance Galerkin, truncation, sampling, and NN-approximation errors, and prescribes how POD ranks and NN parameters should scale to preserve convergence. The methodology is demonstrated on a three-dimensional sound-soft acoustic scattering problem with parametric domain deformations, validating POD rank requirements and NN convergence trends, while highlighting practical training challenges. The results offer a principled pathway to deploy GPOD-NN surrogates in industrially relevant, high-dimensional settings and point to future extensions to time-dependent and electromagnetic scattering problems.

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.

Figures (5)

• Figure 1: NN architecture for the approximation of the map $\boldsymbol{\pi}^{\text{(rb)}}_{h,J}:\mathbb{U} \to \mathbb{C}^J$ with the NN $\boldsymbol{\pi}^{\text{(rb)}}_{\boldsymbol\theta}: \mathbb{U}^{(s)} \to \mathbb{R}^{2J}$ The NN accepts $s\in \mathbb{N}$ inputs in the input layer corresponding to the components of the parametric input $\y = (y_1,\dots,y_s) \in \mathbb{U}^{(s)}$. In addition, there are $2J$ outputs for the approximation of both the real and imaginary parts, i.e. ${\boldsymbol \alpha^\Re_{\boldsymbol \theta}( \boldsymbol{y})}$ and ${\boldsymbol \alpha^\Im_{\boldsymbol \theta}}( \boldsymbol{y})$, respectively, of the reduced coefficients.
• Figure 1: The turbine geometry which is randomly deformed.
• Figure 2: Realizations of the randomly deformed domain. Colors illustrate the real part of the values of the parameter-to-solution map given by \ref{['eq:A_combined']} for the wavenumbers $\kappa=1$ (top) and $\kappa=4$ (bottom).
• Figure 3: Decay of the eigenvalues of the covariance operator \ref{['eq:surfacecovarianceoperator']} used to generate the domain deformations (left) and resulting POD-ranks for wavenumbers $\kappa=1$ and $\kappa=4$ (right).
• Figure 4: Training (top) and generalization errors (upper middle) for the Galerkin-POD neural network approximation. The $L^2(\mathbb{U};L^2(\widehat{\Gamma}))$-error of the POD (lower middle) confirms the theoretical estimates from \ref{['cor:PODrankbound']}. The combined Galerkin-POD NN (bottom) seems to confirm the results from \ref{['cor:NNparamconv']} up to the gap between theory and practice, c.f. \ref{['sec:NN_training']}.

Theorems & Definitions (23)

• Definition 2.1: CCS15
• Remark 1
• Lemma 3.1
• Proof 1
• Corollary 3.2
• Lemma 3.3
• Proof 2
• Corollary 3.4
• Proof 3
• Corollary 3.5