FieldTNN-based machine learning method for Maxwell eigenvalue problems

TL;DR

This work extends the existing TNN-based approach to address the Maxwell eigenvalue problem, a fundamental challenge in electromagnetic field theory, and tackles non-tensor computational domains, which represents a novel and significant contribution of this work.

Abstract

The aim of this paper is to introduce a FieldTNN-based machine learning method for solving the Maxwell eigenvalue problem in both 2D and 3D domains, including both tensor and non-tensor computational regions. First, we extend the existing TNN-based approach to address the Maxwell eigenvalue problem, a fundamental challenge in electromagnetic field theory. Second, we tackle non-tensor computational domains, which represents a novel and significant contribution of this work. Third, we incorporate the divergence-free condition into the optimization process, allowing for the automatic filtering of spurious eigenpairs. Numerical examples are presented to demonstrate the efficiency and accuracy of our algorithm, underscoring its potential for broader applications in computational electromagnetics.

Figures (9)

• Figure 3.1: Architecture of TNN. The red circles $x_{j}~(j=1,\cdots,d)$ are the input data, then the blue circles $\hat{\psi}_{j,k}(x_{j};\theta_{j})~(j=1,\cdots,d,~k=1,\cdots,p)$ are obtain by the hidden layers of each subnetworks (the green cirlces), i.e., the FNN as shown in Eq. \ref{['eq:FNN']}. The orange ovals $\prod\limits_{j=1}^{d} \hat{\psi}_{j, k}\left(x_j ; \theta_{j}\right)~(k=1,\cdots,p)$ are obtained by the scalar multiplication of the blue circles. The summation of $p$ oranges ovals form the final output of FieldTNN.
• Figure 3.2: Architecture of FieldTNN. The red circles $x_{j}~(j=1,\cdots,d)$ are the input data, then the blue squircles $\hat{\psi}_{i,j,k}(x_{j};\theta_{i,j})~(i,j=1,\cdots,d,~k=1,\cdots,p)$ are obtain by the hidden layers each subnetworks (the green cirlces), i.e., the FNN as shown in Eq. \ref{['eq:FNN']}, where $N_{L} = dp$. The orange squircles $\widehat{\Psi}_{i,k}(\mathbf{x}; \Theta_{i})~(i=1,\cdots,d,~k=1,\cdots,p)$ are obtained by the scalar multiplication of the blue ones. Then the purple squircles are derived by vectorized assembly from the orange ones. The summation of $p$ elements in pruple squircles form the final output of TNN.
• Figure 4.1: Illustration for the L-shaped domain.
• Figure 6.1: Maxwell eigenfunctions in square cavity $\Omega = [0,1]^{2}$, exaction solutions (left column), FieldTNN approximation solutions (middle column) and associated absolute errors (right column).
• Figure 6.2: Maxwell eigenfunctions in square cavity $\Omega = [0,1]^{2}$, exaction solutions (left column), FieldTNN approximation solutions (middle column) and associated absolute errors (right column).

Theorems & Definitions (3)

• Remark 4.1
• Remark 4.2: Remark for FieldTNN
• Remark 5.1