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Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators
Han Zhang, Alexander Ogren, Cynthia Rudin, Johann Guilleminot, L. Catherine Brinson
TL;DR
Eigenvalue PDEs require predicting an eigenparameter together with one of several valid eigenmodes, challenging neural operators built around unique input-output maps. The paper uses wavelet-encoded Fourier Neural Operators to select modes deterministically across continuous and binary metamaterial geometries. The surrogate preserves high fidelity while accelerating the simulation stage of metamaterial design by three orders of magnitude relative to finite element analysis.
Problem
Eigenvalue PDEs require simultaneous recovery of an unknown eigenparameter and one of several valid eigenmodes, whereas standard neural operator formulations assume a unique solution map.
Method
The paper augments Fourier Neural Operators with wavelet encodings of modal parameters to learn and deterministically select multiple elastic-wave eigenmodes across continuous and binary metamaterial geometries.
Results
Three orders of magnitude speedup relative to the corresponding FEA eigenvalue solve, while unseen-geometry displacement predictions typically achieve relative errors between 10−2 and 10−4 and reconstructed dispersion relations average below 1% error.
Takeaways & Limitations
Wavelet-encoded FNOs provide a pathway to accelerate repeated Bloch analyses used in metamaterial optimization, uncertainty quantification, inverse design, and geometry screening.
Takeaways & Limitations
Prediction accuracy decreases as geometric discontinuities increase, and robustness across distinct design spaces or symmetry classes remains future work.
Abstract
from arXiv · showhide
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.
1. Introduction
Acoustic metamaterial eigenvalue problems require recovering both an eigenparameter and one of several valid eigenmodes, making them harder for standard neural operators designed for unique solution maps. This work introduces wavelet-encoded FNOs that deterministically select multiple deformation modes while addressing accuracy variation from geometric discontinuities.
- Motivation: Repeated finite element eigenvalue analysis can make metamaterial design cycles prohibitive for optimization, uncertainty quantification, inverse design, and real-time control.Finite element analysis is reliable but expensive when evaluations must be repeated.
- Motivation: Eigenvalue PDEs require simultaneous recovery of an unknown eigenparameter and one of several valid eigenmodes, unlike conventional unique-solution problems.This challenge is especially relevant to metamaterials, where multiple deformation modes correspond to different resonant frequencies and wave behaviors.
- Approach: A single FNO learns multiple eigenmodes across continuous-valued and binary-valued metamaterial geometries, recovering displacement fields and eigenfrequencies.The model targets full Bloch deformation modes rather than only scalar dispersion or transmission quantities.
- Approach: Wavelet encodings of modal parameters enable deterministic mode selection because their spatial–spectral structure is compatible with the FNO architecture.The work contrasts this capability with conventional spatial and purely spectral encodings.
- Findings: Prediction accuracy degrades with geometric discontinuities, which the authors interpret through spectral truncation and its interaction with sharp interfaces.The result motivates examining how discontinuities affect displacement and eigenfrequency recovery.
- Significance: The framework extends neural operators beyond conventional forward PDE problems and offers a pathway to accelerate repeated finite element eigenvalue analysis in metamaterial design.The stated applications include optimization, uncertainty quantification, inverse design, and screening of unit-cell geometries.
2. Methodology
The methodology combines Bloch FEA data generation with an FNO that predicts multiple acoustic metamaterial eigenmodes from geometry and wavelet-encoded modal parameters. The study evaluates continuous and binary geometries and uses wavelet encodings to preserve modal information in both spatial and spectral representations.
- Metamaterial design space: The design space comprises two-dimensional, eight-fold-symmetric square unit cells discretized on 32 × 32 grids with steel and stiff-elastomer material properties.Material channels represent normalized elastic modulus, density, and Poisson’s ratio, while continuous and thresholded binary geometries each comprise half the dataset.
- Metamaterial design space: Continuous geometries are generated from periodic Gaussian-process samples, while binary geometries threshold those samples into pure elastomer and pure steel regions.The continuous field is clipped to [0, 1] and constrained to eight-fold symmetry before binary thresholding is applied.
- Wave propagation simulations: Bloch FEA solves a generalized eigenvalue problem for each geometry and wavevector, producing complex displacement eigenvectors and angular eigenfrequencies.Geometry-dependent stiffness and mass matrices are reduced using a wavevector-dependent Bloch transformation; the IBZ contains 325 wavevectors and six computed bands per wavevector.
- Dataset construction: The FEA targets contain four real-valued displacement channels plus a spatially uniform frequency channel, yielding a 5 × 32 × 32 output tensor.The FNO input combines the geometry channel with wavelet encodings of the wavevector and band index rather than directly ingesting the three FEA property channels.
- Fourier neural operator: The FNO expands inputs into hidden channels, alternates pointwise and Fourier spectral branches, and projects the four-layer latent representation to five outputs.The spectral branch applies an FFT, learned weights to retained modes, and an inverse FFT; the selected architecture uses 128 hidden channels and GELU activations.
- Wavelet encoding and training: Wavelet encodings make modal parameters spectrally distinguishable, with a maximum off-diagonal similarity of 0.862 across the 325 wavevector encodings.The study reports lower test losses and better-resolved displacement fields than constant-field or sinusoidal alternatives, while deeper or wider models increased cost and overfitting without appreciable validation gains.
3. Results and Discussion
The FNO recovers displacement fields and eigenfrequencies across continuous and binary geometries, with wavelet encodings enabling deterministic multi-mode prediction. Accuracy is generally high but degrades for binary geometries with greater interface complexity and for encodings that underuse the Fourier branch.
- Displacement predictions: The evaluation compares displacement-field predictions across unseen continuous and binary geometries using NMAE and performance percentiles.Representative cases use the 25th, 50th, and 75th performance percentiles for both geometry datasets.
- Continuous geometries: For continuous geometries, average pixel relative errors decrease from approximately 10^-2 to 10^-4 as performance percentile increases.Upper-percentile predictions match target structure and scale closely, with the 75th-percentile case virtually indistinguishable by eye.
- Multi-mode learning: A single wavelet-encoded FNO learns multiple deformation modes and performs across both continuous and binary geometries without separate specialization.The results support shared structure in the underlying eigenvalue map and deterministic mode selection.
- Geometric complexity: Binary geometries are systematically harder because jump discontinuities and broadband high-wavenumber content are less faithfully resolved by truncated Fourier representations.Prediction difficulty increases with sharper material transitions and longer interfaces on the fixed grid.
- Eigenfrequency reconstruction: The model achieves average prediction error below 1% on the encoded eigenfrequency across unseen samples.Here, encoded eigenfrequency denotes a log-transformed, spatially uniform output channel rather than physical frequency in Hz.
- Geometric complexity: Boundary length has a positive monotonic association with MAE across every band, with Spearman correlations from ρ = 0.644 for band 1 to approximately ρ = 0.42–0.56 for higher bands.Mean wavevector-averaged MAE also rises with band index, from approximately 1.6 × 10^-3 for band 1.
- Encoding comparison: Wavelet encodings outperform sinusoidal alternatives because they populate both spatial and spectral structure used by the FNO.Sinusoidal encodings leave the Fourier branch weakly utilized, producing persistent graininess that increased model depth and width did not remove.
4. Conclusions
A single FNO with wavelet encodings learns multiple metamaterial eigenmodes across continuous and binary geometries, while achieving high-fidelity predictions and substantial speedups over finite element analysis. Accuracy declines with increasing geometric discontinuity, but the surrogate remains practical for repeated design analyses.
- A single FNO recovers Bloch displacement fields and eigenfrequencies across continuous and binary unit cell geometries.Wavelet encodings enable deterministic selection among multiple modes.
- 10^-2 to 10^-4 relative displacement-field errors and below 1% average dispersion-relation error are achieved on unseen geometries.
- Constant encodings fail to propagate information through spectral FNO branches, whereas sinusoidal encodings train better but remain grainy.
- Gabor wavelet encodings provide richer information in both spatial and spectral branches, enabling more reliable mode selection and learning.
- Prediction error grows with input and output discontinuities, as performance decreases with increasing interface length.
- 1 ms per sample versus about 1 s for the corresponding FEA eigenvalue solve yields a three-orders-of-magnitude speedup.The trained surrogate is about 2 GB at float16 and runs on a consumer-grade CPU.
- The speedup makes repeated Bloch analyses more practical for optimization, uncertainty quantification, inverse design, and geometry screening.
AI Usage Disclosure
Generative AI tools assisted with coding, code translation, training-pipeline automation, and light manuscript editing; the authors reviewed and approved the scientific content and final wording.
- Generative AI tools assisted with coding experiments, translating MATLAB code to Python, and automating training pipelines.
- AI tools also provided light editing of the manuscript.
- The authors reviewed and approved all scientific claims, results, and final wording.
CRediT authorship contribution statement
The contribution statement assigns research, software, supervision, writing, and project responsibilities across the listed authors.
- Han Zhang contributed conceptualization, data curation, formal analysis, investigation, methodology, software, validation, visualization, and manuscript writing.
- Alexander Ogren contributed data curation and software.
- Cynthia Rudin contributed supervision and manuscript review and editing.
- Johann Guilleminot contributed formal analysis, methodology, supervision, and manuscript review and editing.
- L. Catherine Brinson contributed conceptualization, funding acquisition, methodology, project administration, resources, software, supervision, visualization, and manuscript review and editing.
S1.1. Model and Training Hyperparameters
Table S1 lists the hyperparameter combinations tested in a grid search for model and training optimization.
- Table S1 reports combinations tried in a grid search for optimal model and training hyperparameters.Each row can be paired with any row from the other sets.
S1.2. Loss Criterion
The study compares several loss functions for signed-valued field reconstruction and selects NMAE for all reported results because it balances accuracy across channels with different magnitudes.
- Loss selection: NMAE yielded the best overall performance and was used for all reported results.It was preferred to MAE, MSE, NMSE, and SSIM.
- Loss behavior: MAE preserved sharper interfaces and boundary features than MSE but underperformed NMAE on overall validation accuracy.Its linear penalty avoids disproportionately penalizing isolated large residuals near discontinuities.
- Loss behavior: MSE reproduced solution magnitude and large-scale structure but produced diffuse interfaces, blurred boundaries, and grainy artifacts near fine-scale features.Its quadratic penalty increasingly weights large residuals and favors smooth predictions.
- Loss limitations: SSIM can assign nearly the same score to a signed field and its sign-reversed counterpart, allowing physically incorrect polarity inversions.The authors therefore advise supplementing SSIM with a loss that explicitly penalizes sign reversals rather than using it alone.
- Normalized losses: NMAE emphasizes relative error, improving reconstruction of low-amplitude displacement components without sacrificing usable accuracy on larger-magnitude channels.This makes it suitable for outputs whose channels have disparate magnitudes.
S1.3. Algorithms for Input Wavelet Encoding
The input encodings use Gabor wavelets to represent band indices and continuous wavevectors in spatial and spectral forms matched to the FNO grid.
- Encoding design: The band and wavevector conditioning channels are generated by Gabor wavelet embeddings matched to the accompanying implementation.Together, they provide the FNO with the conditioning channels used for the model.
- Band-index encoding: The band-index embedding uses a 32 × 32 grid, with successive bands increasing carrier frequency and orientation.Orientation repeats every eight bands, while frequency continues increasing so embeddings remain distinguishable beyond eight bands.
- Wavevector encoding: The wavevector embedding uses continuous radian-valued kx and ky components with distinct modulo periods Nx = 13 and Ny = 25.These choices reduce aliasing when the two wavevector components are interchanged.
- Wavevector encoding: The wavevector embedding maps physical k values into resolved carrier frequencies while keeping patterns spatially localized and informative in both domains.The offset f0, scale α, and envelope width σ = 0.5/r control this mapping and localization.
S1.4. Wavelet Decoding Fidelity
The scalar encoding places values on a logarithmic scale in Gabor-like images, while decoding uses Fourier-spectrum peaks to recover the value across a wide range.
- Fidelity experiment: The scalar round-trip experiment tests recovery over [1, 8000], spanning about four orders of magnitude.The encoding and decoding procedures are summarized by Algorithms 3 and 4.
- Fidelity experiment: Decoded values remain accurate across the full range despite finite-grid discretization and float16 precision limits.Figures S1 and S2 show representative encodings and corresponding round-trip error.
- Fidelity visualization: The encodings alternate spatial wavelet images with their corresponding Fourier magnitude spectra across logarithmically spaced scalar values.This visualization covers scalars from 1 to 8000.
- Encoding: The forward mapping jointly encodes scalar magnitude and orientation into a log-scaled Gabor-like pattern.The scalar is embedded as a positive value using the specified bounds and wavelet parameters.
- Decoding: The inverse mapping detects a dominant Fourier-magnitude peak, refines its centroid, and combines magnitude and angle to reconstruct the most likely log value.It tests neighboring bands for a consistent decoded value.