Embedding a de Rham Complex Structure into Eig-PIELM for Maxwell Eigenvalue Problems
Abstract
The perfect electric conductor (PEC) boundary condition alone does not eliminate the gradient nullspace of the Maxwell curl-curl operator. Consequently, a boundary-only Maxwell implementation of Eig-PIELM, an eigenvalue extension of the physics-informed extreme learning machine (PIELM), may retain represented gradient modes in its reduced trial space, producing nonphysical, gradient-contaminated modes. We develop dR-Eig-PIELM, a finite-dimensional structure-preserving extension that retains the linear-coefficient, boundary-null-space, and one-shot multimode architecture of Eig-PIELM. It combines compatible tensor-product grad--curl trial spaces with a constrained null-space construction to restrict the sampled PEC space to the mass-orthogonal complement of the represented gradient image before the eigensolve. Symmetric weak-form Gram assembly yields a symmetric-definite reduced eigenproblem without an eigenvalue cutoff under the stated sampling and material-positivity assumptions. For rectangles and cuboids, we establish fixed-degree exactness at the PEC vector-field space and derive rank and dimension identities. Numerical experiments show that dR-Eig-PIELM excludes the spurious eigenmodes with significant gradient components observed in the boundary-only baseline. The method exhibits rapid convergence with increasing polynomial degree and competitive work--precision performance in the tested configurations. A covariantly mapped waveguide example illustrates the potential of dR-Eig-PIELM for integration into design optimization workflows through a constrained local geometry update driven by computed cutoff frequencies and their sensitivities. This work provides a mathematical and computational framework for separating represented gradient modes from physical Maxwell eigenmodes within Eig-PIELM and supports its use in waveguide spectral analysis and geometry design.
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BibTeX
@misc{kang2026embedding,
title = {{Embedding a de Rham Complex Structure into Eig-PIELM for Maxwell Eigenvalue Problems}},
author = {Hao Kang and Liang Li and Xing Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02337},
eprint = {2611.02337},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02337},
}
RIS
TY - UNPB
TI - Embedding a de Rham Complex Structure into Eig-PIELM for Maxwell Eigenvalue Problems
AU - Hao Kang
AU - Liang Li
AU - Xing Li
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02337
UR - https://arxiv.org/abs/2611.02337
AB - The perfect electric conductor (PEC) boundary condition alone does not eliminate the gradient nullspace of the Maxwell curl-curl operator. Consequently, a boundary-only Maxwell implementation of Eig-PIELM, an eigenvalue extension of the physics-informed extreme learning machine (PIELM), may retain represented gradient modes in its reduced trial space, producing nonphysical, gradient-contaminated modes. We develop dR-Eig-PIELM, a finite-dimensional structure-preserving extension that retains the linear-coefficient, boundary-null-space, and one-shot multimode architecture of Eig-PIELM. It combines compatible tensor-product grad--curl trial spaces with a constrained null-space construction to restrict the sampled PEC space to the mass-orthogonal complement of the represented gradient image before the eigensolve. Symmetric weak-form Gram assembly yields a symmetric-definite reduced eigenproblem without an eigenvalue cutoff under the stated sampling and material-positivity assumptions. For rectangles and cuboids, we establish fixed-degree exactness at the PEC vector-field space and derive rank and dimension identities. Numerical experiments show that dR-Eig-PIELM excludes the spurious eigenmodes with significant gradient components observed in the boundary-only baseline. The method exhibits rapid convergence with increasing polynomial degree and competitive work--precision performance in the tested configurations. A covariantly mapped waveguide example illustrates the potential of dR-Eig-PIELM for integration into design optimization workflows through a constrained local geometry update driven by computed cutoff frequencies and their sensitivities. This work provides a mathematical and computational framework for separating represented gradient modes from physical Maxwell eigenmodes within Eig-PIELM and supports its use in waveguide spectral analysis and geometry design.
ER -