Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators
Abstract
Machine learning surrogates based on neural operators have shown broad applicability in solving forward PDE problems. However, eigenvalue problems, in which an eigenparameter and one of several valid eigenmodes must be simultaneously solved, remain difficult because standard operator learning formulations assume a unique input-output map. This work demonstrates that Fourier Neural Operators (FNOs), combined with wavelet-based encodings of PDE inputs, can learn and predict multiple eigenmodes of the elastic wave equation, corresponding to deformation modes of acoustic waves propagating through arbitrary metamaterial geometries. We provide a mechanistic explanation and experimental evidence for why wavelet encodings are well matched to the dual spatial-spectral structure of the FNO, enabling deterministic mode selection on both continuous-valued and binary-valued geometries within a single model, and for why prediction accuracy varies with geometric discontinuities. For metamaterial design, the resulting surrogate accelerates the simulation stage of the design cycle by three orders of magnitude relative to finite element analysis on a consumer-grade CPU, while preserving high fidelity. These results also carry broader implications for designing input encodings in other multi-mode PDE solvers based on spectral neural operators.
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BibTeX
@misc{zhang2026learning,
title = {{Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators}},
author = {Han Zhang and Alexander Ogren and Cynthia Rudin and Johann Guilleminot and L. Catherine Brinson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08102},
eprint = {2609.08102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08102},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators
AU - Han Zhang
AU - Alexander Ogren
AU - Cynthia Rudin
AU - Johann Guilleminot
AU - L. Catherine Brinson
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08102
UR - https://arxiv.org/abs/2609.08102
AB - Machine learning surrogates based on neural operators have shown broad applicability in solving forward PDE problems. However, eigenvalue problems, in which an eigenparameter and one of several valid eigenmodes must be simultaneously solved, remain difficult because standard operator learning formulations assume a unique input-output map. This work demonstrates that Fourier Neural Operators (FNOs), combined with wavelet-based encodings of PDE inputs, can learn and predict multiple eigenmodes of the elastic wave equation, corresponding to deformation modes of acoustic waves propagating through arbitrary metamaterial geometries. We provide a mechanistic explanation and experimental evidence for why wavelet encodings are well matched to the dual spatial-spectral structure of the FNO, enabling deterministic mode selection on both continuous-valued and binary-valued geometries within a single model, and for why prediction accuracy varies with geometric discontinuities. For metamaterial design, the resulting surrogate accelerates the simulation stage of the design cycle by three orders of magnitude relative to finite element analysis on a consumer-grade CPU, while preserving high fidelity. These results also carry broader implications for designing input encodings in other multi-mode PDE solvers based on spectral neural operators.
ER -