Galerkin Neural Network-POD for Acoustic and Electromagnetic Wave Propagation in Parametric Domains

TL;DR

This paper introduces a Galerkin POD neural network approach for efficiently solving parametric acoustic and electromagnetic wave problems, enabling rapid online evaluations with high accuracy in complex domains.

Contribution

It combines reduced basis methods with neural networks to decouple offline training from online solution evaluation for parametric wave problems.

Findings

Significant online speed-up over traditional methods.

Accurate solutions for 3D Helmholtz and Maxwell equations.

Effective handling of high-dimensional parametric domains.

Abstract

We investigate reduced-order models for acoustic and electromagnetic wave problems in parametrically defined domains. The parameter-to-solution maps are approximated following the so-called Galerkin POD-NN method, which combines the construction of a reduced basis via proper orthogonal decomposition (POD) with neural networks (NNs). As opposed to the standard reduced basis method, this approach allows for the swift and efficient evaluation of reduced-order solutions for any given parametric input. As is customary in the analysis of problems in random or parametrically defined domains, we start by transporting the formulation to a reference domain. This yields a parameter-dependent variational problem set on parameter-independent functional spaces. In particular, we consider affine-parametric domain transformations characterized by a high-dimensional, possibly countably infinite, parametric input. To keep the number of evaluations of the high-fidelity solutions manageable, we propose using low-discrepancy sequences to sample the parameter space efficiently. Then, we train an NN to learn the coefficients in the reduced representation. This approach completely decouples the offline and online stages of the reduced basis paradigm. Numerical results for the three-dimensional Helmholtz and Maxwell equations confirm the method's accuracy up to a certain barrier and show significant gains in online speed-up compared to the traditional Galerkin POD method.