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

TL;DR

This work develops a Galerkin POD-NN reduced-order framework for 3D acoustic and electromagnetic wave propagation in domains with affine-parametric deformations. By combining POD-based reduced bases with neural networks that learn the parameter-to-coefficient map, the approach achieves non-intrusive offline-online decoupling and substantial online speedups relative to classical Galerkin POD. The authors establish parametric holomorphy for the Helmholtz impedance and Maxwell lossy cavity problems, derive convergence rates that are dimension-independent, and handle complex-valued solutions by separating real and imaginary parts in the NN. Numerical experiments on Helmholtz and Maxwell models demonstrate the method’s accuracy and propose centered RB-POD as a practical enhancement, while also discussing data requirements, sampling strategies, and potential extensions to multi-fidelity learning and active data acquisition.

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.

Figures (16)

• Figure 1: NN architecture for the approximation of the map $\boldsymbol{\pi}^{\text{(rb)}}_{L,\mathbb{R}}: \mathrm{U}^{(J)} \to \mathbb{R}^{2L}$ as in \ref{['eq:ann_parametric_real']}. The NN accepts as input $J$ values accounting for the components of the parametric input $\y = (y_1,\dots,y_J) \in \mathrm{U}^{(J)}$, whereas there are $2L$ outputs representing both the real and imaginary parts of the reduced coefficients. The input (red) and hidden layers (blue) are fully connected (FC) with hyperbolic tangent ($\tanh$) activation functions.
• Figure 2: Computational meshes and graphical results for the Helmholtz problem: (A) reference mesh, on which the solutions are computed. (B) Physical domain. (C) Imaginary part of the full-order solution. (D) Imaginary part of the POD-NN prediction. The solution to the Helmholtz problem was computed for the parameters $\theta = 0.5, l=0.1, \nu=0.5$, and $J =50$. The domain deformation is amplified by a factor of two for better visibility.
• Figure 3: The first 50 coefficients $\mu_j$ for different parameters of algebraic (blue tones) and Matérn (red tones) decay.
• Figure 4: The singular values of 1024 snapshots for different parameters of algebraic (blue tones) and Matérn (red tones) decay. The input parameters originate from the same Halton sequence.
• Figure 5: Test errors for different neural network architectures, i.e., different numbers of hidden layers $D$ and neurons per layer $H$. All models were trained using Adam with learning rate 5e-4, $\beta_1 = 0.8$ and $\beta_2 = 0.9$ for 4000 epochs. The networks were trained on 1024 snapshots sampled from a Halton sequence of Matérn decay parameters with the following settings: $\theta = 0.1, J = 50, \nu = 0.5, l = 0.1$.

Theorems & Definitions (11)

• Remark 2.2
• Definition 3.1: CCS15
• Proposition 3.2: Parametric Holomorphy of the Discrete Parameter-to-Solution Map
• proof
• Lemma 3.3: Decay of Kolmogorov's Width, CD16
• Theorem 3.4: Convergence of the Galerkin-POD RB Method
• proof
• Lemma 4.1
• proof
• Lemma 4.2