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Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators

Han Zhang, Alexander Ogren, Cynthia Rudin, Johann Guilleminot, L. Catherine Brinson

arXiv:2609.08102v1cs.LGcond-mat.mtrl-scics.CEphysics.comp-ph

TL;DR

Eigenvalue problems require solving an eigenparameter and choosing among valid modes, a challenge for operator learning framed as a unique input-output map. The paper combines FNOs with wavelet encodings to predict metamaterial eigenmodes, reporting high-fidelity predictions and a three-orders-of-magnitude simulation speedup over finite element analysis.

  • Problem

    Metamaterial design requires computing multiple deformation modes, while repeated simulation can make design-cycle evaluation prohibitive.

    Source: §1 · p. 3

  • Method

    The paper combines FNOs with wavelet encodings of modal parameters to learn multiple eigenmodes and deterministically select modes.

    Source: §1 · p. 4

  • Results

    On unseen geometries, displacement predictions typically achieve relative errors between 10^-2 and 10^-4, and reconstructed dispersion relations average below 1% error.

    Source: §4 · p. 33

  • Takeaways & Limitations

    Three orders of magnitude speedup over corresponding FEA eigenvalue solves makes repeated Bloch analyses for design iterations more practical.

    Source: §4 · p. 34

    Accuracy tends to degrade as binary designs become more discontinuous, and robustness across distinct design spaces or symmetry classes remains untested.

    Source: §3.1.3 · p. 24

Abstract

from arXiv · show

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

This work extends Fourier Neural Operators to elastic-wave eigenvalue problems in acoustic metamaterials, using wavelet encodings to select among multiple valid modes. It evaluates recovery across continuous- and binary-valued geometries and relates accuracy loss at geometric discontinuities to spectral truncation.

1. Introduction

  • Repeated finite-element eigenvalue analysis can make simulation prohibitive for optimization, uncertainty quantification, inverse design, and real-time control.The framework offers a pathway to accelerate metamaterial simulation and design relative to repeated finite-element analysis.Source: §1 · pp. 3, 5
  • Eigenvalue PDEs require simultaneous recovery of an unknown eigenparameter and one of several valid modes, unlike PDE mappings with unique solutions.Predicting full Bloch displacement eigenfields with deterministic multi-mode selection using an FNO is identified as an unsolved challenge.Source: §1 · pp. 3–4
  • A single FNO learns multiple eigenmodes across continuous- and binary-valued metamaterial geometries, recovering displacement fields and eigenfrequencies.The model reproduces distinct modes deterministically on demand.Source: §1 · p. 4
  • Wavelet encodings of modal parameters enable deterministic mode selection by matching the spatial–spectral structure of FNOs.Conventional spatial and purely spectral encodings do not provide the same capability.Source: §1 · p. 4
  • Prediction accuracy degrades with geometric discontinuities, which the work interprets through FNO spectral truncation.This interpretation bears on applying FNOs to problems with sharp interfaces.Source: §1 · pp. 4–5

2. Methodology

The methodology develops a supervised FNO surrogate using Bloch finite-element simulations, with wavelet encodings of wavevector and band conditioning predictions across continuous and binary metamaterial geometries. It describes the geometry and embedding inputs, displacement and eigenfrequency outputs, and the model’s training and validation setup.

2.1. Metamaterial Design Space

  • The design space uses eight-fold-symmetric, 32 × 32 square unit cells with three normalized material-property channels and continuous or thresholded binary geometries.The two geometry representations each make up half of the dataset.Source: §2.1 · pp. 5–6

2.2. Wave Propagation Simulations

  • Bloch–Floquet FEA generates supervised labels by solving for six lowest eigenfrequencies and displacement modes at 325 wavevectors per geometry.Complex displacement components become four real-valued channels, with frequency as a fifth target channel.Source: §2.2 · pp. 7–8

2.3. Fourier Neural Operator

  • An FNO maps a geometry channel plus wavelet-encoded wavevector and band to predicted frequency and displacement fields through lifting, Fourier layers, and projection.Its spectral convolutions capture long-range interactions and periodic structure, while pointwise transformations preserve local geometric information.Source: §2.3 · pp. 10–11

2.4. Input Wavelet Encoding

  • Across 325 IBZ wavevector encodings, the maximum off-diagonal spectral cosine similarity was 0.862, indicating distinct wavevector pairs remain spectrally separable on the 32 × 32 grid.Source: Figure 6 · p. 14

2.5. Dataset Construction

  • The dataset combines 24,000 continuous and binary geometries in one training pool, yielding 46.8 million geometry–wavevector–band input-output pairs.Each pair maps a 3 × 32 × 32 input to a 5 × 32 × 32 output.Source: §2.5 · p. 15
  • Training samples paired 3 × 32 × 32 geometry-and-wavelet input tensors with 5 × 32 × 32 outputs for eigenfrequency and displacement components.The five output channels are eigenfrequency, then real and imaginary components of x and y displacements.Source: §2.6 · pp. 15–16, Figure 7 · p. 15

2.6. Training Procedure

  • NMAE gave the best aggregate performance among five evaluated loss functions and was used for all reported results.Losses were aggregated across all five output channels.Source: §2.6 · pp. 15–16
  • AdamW outperformed Adam, other momentum-based optimizers, and SGD; the final configuration used a 2 × 10^-3 learning rate, batch size 520, and 12 epochs.Learning rates above approximately 10^-2 occasionally caused an abrupt, irreversible loss spike followed by a training plateau.Source: §2.6 · p. 16
  • The selected architecture used four Fourier layers, 128 hidden channels, and GELU; greater depth or width did not appreciably improve validation performance and increased cost and overfitting.Final performance also did not appreciably change with the scheduler step size or batch-size variations tested.Source: §2.6 · p. 16

3. Results and Discussion

The section examines how wavelet-based conditioning supports FNO prediction of displacement fields and eigenfrequencies across continuous and binary geometries, including how prediction difficulty varies with geometry discontinuity and eigenband. It also compares wavelet, sinusoidal, and uniform encodings in relation to prediction accuracy and deterministic mode selection.

3.1. Displacement Field Prediction

  • The displacement evaluation measures NMAE over four real-valued channels and compares representative unseen samples at the 25th, 50th, and 75th performance percentiles for both geometry datasets.The percentiles represent relatively weak, median, and relatively strong predictions, respectively.Source: §3.1 · p. 17

3.1.1. Continuous Geometries

  • On continuous geometries, wavelet-encoded predictions reproduce displacement structure and scale well, with target and prediction virtually indistinguishable by eye at the 75th performance percentile.Even at the 25th percentile, outcomes are described as fairly good; average pixel relative errors decrease from around 10^-2 to 10^-4 as performance percentile increases.Source: §3.1.1 · pp. 18, 20

3.1.2. Binary Geometries

  • On binary geometries, predictions show strong structural and scale agreement at the median and are hard to distinguish by eye at the 75th percentile, but low-amplitude channels can retain large relative errors.NMAE weights channels relative to target magnitude; in the cited sample, imaginary channels are one to two orders of magnitude larger than real channels, so absolute error can remain low while relative error on smaller channels is large.Source: §3.1.2 · pp. 21–23

3.1.3. Continuous Versus Binary Performance

  • With wavelet encodings, the same FNO performs well on continuous and binary geometries; average pixel relative errors in upper performance quartiles decrease from on the order of 10^-2 to 10^-4 as performance percentile increases.Source: §3.1.1 · p. 20, §3.1.2 · p. 23, §3.1.3 · p. 24

3.2. Dispersion Band Reconstruction

  • Across unseen continuous and binary geometries, average prediction error on the encoded eigenfrequency is below 1%.The eigenfrequency output is log-transformed and spatially uniform, rather than physical frequency in Hz; the plotted dispersion bands cover representative cases.Source: Figure 15 · p. 25, Figure 16 · p. 26, §3.2 · p. 26

3.3. Dependence of Prediction Error on Band and Boundary Length

  • For binary designs, interface length is positively associated with displacement MAE in every band, while mean wavevector-averaged MAE rises from 1.6 × 10^-3 in band 1 to 3.8 × 10^-3 in band 6.Spearman correlations range from ρ = 0.644 for band 1 to approximately 0.42–0.56 for higher bands; the findings link error to both geometric and modal complexity.Source: Figure 18 · p. 29, §3.3 · pp. 29–30

3.4. Band and Wavevector Encoding

  • Constant-field conditioning leaves the FNO Fourier branch with little nontrivial information beyond the first layer, and can collapse to near-constant outputs that ignore conditioning under some settings.With more stable settings, the constant encoding still underperforms wavelet encoding.Source: §3.4 · pp. 30–31
  • The sinusoidal wavevector field is a directed plane wave whose phase gradient follows the physical wavevector, with spatial frequency scaling with |k|.Source: §3.4 · p. 31
  • Sinusoidal encoding concentrates energy at few wavenumbers, weakly utilizes the Fourier branch, and produces grainy, under-resolved predictions that greater model depth and width do not fix.This indicates that the bottleneck is the encoding rather than model capacity.Source: §3.4 · p. 31
  • Wavelet encoding yields accurate, well-resolved predictions and deterministic mode selection by supplying usable structure to both branches of each Fourier layer.Gabor wavelets trade certainty between spatial and spectral domains, keeping both informationally rich.Source: §3.4 · p. 31
  • The results support matching input encodings to the operator’s spatial and spectral structure: wavelet encodings exercise both domains, whereas single-domain encodings underuse the FNO’s Fourier branch.Source: §3.4 · pp. 31–32
  • The evaluation compares wavelet-, sinusoidal-, and uniform-encoded models on held-out continuous and binary geometries using overall-sample MAE, MSE, NMAE, and NMSE under matched training conditions.The reported checkpoints follow eight epochs for an equal training-budget comparison; Tables 2 and 3 describe median and mean test losses, respectively.Source: §3.4 · p. 32, Table 2 · p. 32, Table 3 · p. 33

4. Conclusions

A single Fourier Neural Operator with wavelet encodings learns multiple elastic-wave eigenmodes across continuous and binary metamaterial geometries. The surrogate combines high-fidelity predictions with substantially faster evaluation than finite element eigenvalue analysis, while performance declines as input and output discontinuities increase.

Conclusions

  • On unseen continuous and binary geometries, the FNO predicts full Bloch displacement fields and eigenfrequencies, with displacement relative errors of 10−2 to 10−4 and average dispersion-relation error below 1%.Unlike prior surrogates mainly targeting dispersion curves, transmission spectra, or bandgap scalars, it deterministically selects among modes using wavelet encodings of wavevector and band index.Source: §4 · p. 33
  • Gabor wavelet encodings provide information to both spatial and spectral FNO branches, enabling more reliable mode selection and learning than constant or sinusoidal encodings.Constant encodings fail to propagate information through spectral branches, while sinusoidal encodings train better but produce grainy predictions because information is unbalanced between branches.Source: §4 · p. 33
  • FNO prediction error tends to increase with input and output discontinuities, with performance decreasing as interface length increases, although the trained surrogate remains high fidelity.The conclusion identifies geometric discontinuities as a limitation on prediction accuracy.Source: §4 · p. 33
  • About 1 ms per sample versus about 1 s for the corresponding FEA eigenvalue solve yields a three-orders-of-magnitude speedup; the float16 surrogate is about 2 GB.This makes repeated Bloch analyses for optimization, uncertainty quantification, inverse design, and geometry screening more practical than direct finite element evaluation during design iterations.Source: §4 · p. 34
  • Future work includes testing other spatial resolutions, physics-informed operator losses, and broader design spaces beyond eight-fold-symmetric, two-material unit cells.Proposed geometries include other tiling symmetries, asymmetric geometries, and multimaterial compositions.Source: §4 · p. 34

AI Usage Disclosure

Generative AI tools assisted with experiment coding, manuscript editing, and training-pipeline automation; the authors reviewed and approved all scientific claims, results, and final wording.

  • AI tools assisted with experiment coding, MATLAB-to-Python translation, training-pipeline automation, and light manuscript editing; authors reviewed and approved all scientific claims, results, and final wording.Source: AI Usage Disclosure · p. 34

CRediT authorship contribution statement

The authors’ contributions span conceptualization, data curation, analysis, methodology, software, supervision, and writing, with roles distributed across five contributors.

CRediT authorship contribution statement

S1. Supplementary Information · S1.1. Model and Training Hyperparameters

The supplementary section describes a grid search over model and training hyperparameters, with each listed set combinable with sets from other rows.

S1.1. Model and Training Hyperparameters

  • Table S1 lists hyperparameter combinations tried in the grid search, with each row’s set pairable with sets from other rows.The search aimed to identify optimal model and training hyperparameters.Source: Table S1 · p. 39

S1.2. Loss Criterion

NMAE was selected for all reported results because it provided the best overall performance, balancing errors across channels with disparate magnitudes. The alternatives exposed trade-offs: MAE preserved sharper boundaries, MSE blurred them, SSIM could reward polarity-inverted fields, and NMSE underperformed NMAE on validation accuracy.

S1.2. Loss Criterion

  • NMAE gave the best overall performance and was selected for all reported results over MAE, MSE, NMSE, and SSIM.By emphasizing relative error, it improved reconstruction of low-amplitude displacement components without sacrificing usable accuracy on larger-magnitude channels.Source: S1.2. Loss Criterion · pp. 39, 42
  • MAE produced sharper interfaces and better-preserved boundary features than MSE, but underperformed NMAE on overall validation accuracy.Its linear penalty does not disproportionately penalize isolated large residuals near discontinuities, but MAE alone was not selected for reported models.Source: S1.2. Loss Criterion · pp. 39–40
  • MSE reproduced solution magnitude and large-scale structure but produced diffuse interfaces, blurred boundaries, and grainy artifacts in place of fine-scale features.Its quadratic penalty prioritizes larger residuals, favoring smooth predictions over sharp transitions.Source: S1.2. Loss Criterion · p. 40
  • For signed-valued fields, SSIM can assign nearly the same score to a perfect reconstruction and its sign-reversed counterpart, allowing physically incorrect polarity inversions.Predictions could retain accurate interfaces and morphology yet invert localized regions or the entire field; SSIM should therefore not be used alone unless another loss penalizes sign reversals.Source: S1.2. Loss Criterion · pp. 40–42
  • NMSE reduced relative error in low-amplitude regions compared with MSE, but underperformed NMAE on overall validation accuracy and was not selected.NMSE combines target-magnitude normalization with MSE's quadratic penalty on larger residuals.Source: S1.2. Loss Criterion · p. 43

S1.3. Algorithms for Input Wavelet Encoding

The section specifies two Gabor wavelet encodings for FNO conditioning: one generated from band indices and one from continuous wavevector components.

Algorithm 1: 1D Gabor Wavelet Embedding from Band Index

  • With S = 32, band embeddings use f = (1 + |b|)S/8, so frequency rises with band index and patterns remain distinguishable beyond eight bands.The orientation repeats every eight bands, while carrier frequency continues to increase; r = 2.0, S/16, and 0.3 are chosen to keep wavelets spatially broad and visually distinct.Source: S1.3. Algorithms for Input Wavelet… · p. 44

Algorithm 2: 2D Gabor Wavelet Embedding for Wavevectors

  • For wavevectors, continuous radian-valued kx and ky use distinct modulo periods Nx = 13 and Ny = 25 to reduce aliasing when the components are interchanged.The offset f0 = 2.20 and scale α = 41 map physical wavevector values to frequencies resolved on the S × S grid; σ = 0.5/r localizes patterns while keeping them informative in both domains.Source: S1.3. Algorithms for Input Wavelet… · pp. 45–46

S1.4. Wavelet Decoding Fidelity

A Gabor-wavelet scalar encoding and spectral decoding experiment tests whether positive scalars survive pixelization, storage, and inference, finding accurate recovery across the tested range despite finite-grid discretization and float16 precision.

S1.4. Wavelet Decoding Fidelity

  • Decoding uses spectral peak detection with centroid refinement, then combines extracted magnitude and angle to reconstruct the most likely log value.The reported encode–decode pair is implemented in Algorithms 3 and 4.Source: S1.4. Wavelet Decoding Fidelity · pp. 46, 48
  • Across [1, 8000], decoded scalars remain accurate despite finite-grid discretization and float16 storage and arithmetic.Figure S2 presents encode–decode error over this range; no specific error values are reported in the supplied passages.Source: S1.4. Wavelet Decoding Fidelity · p. 46, Figure S2 · p. 48
  • The forward mapping places each positive scalar on a log scale and jointly encodes magnitude and orientation in a Gabor-like pattern.The experiment uses a 32 × 32 image and tests scalars spanning about four orders of magnitude.Source: S1.4. Wavelet Decoding Fidelity · pp. 46, 48, 49

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