Fourier neural operators learn multiple acoustic wave patterns in metamaterials
Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators
Machine LearningComputational Engineering, Finance, and Science
Summary
Simulating how sound waves vibrate and move through complex materials called metamaterials can be very slow using traditional methods. The authors show that a type of machine learning model, called a Fourier Neural Operator combined with wavelet encodings, can quickly and accurately predict multiple patterns of wave vibrations in these materials. This method helps select specific wave patterns and works well for different shapes, even with sharp edges. It speeds up simulations by about a thousand times on a regular computer while still being precise. Their approach also suggests ways to improve similar machine learning tools for other physics problems.
Fourier neural operatorwavelet encodingeigenvalue problemmetamaterialselastic wave equationeigenmodepartial differential equationsspectral methodsfinite element analysismachine learning surrogate
Authors
Han Zhang, Alexander Ogren, Cynthia Rudin, Johann Guilleminot, L. Catherine Brinson
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.