学习超材料本征模式：基于小波编码的傅里叶神经算子

中文摘要 AI 辅助

基于神经算子的机器学习代理模型在求解正向偏微分方程问题中展现出广泛的适用性。然而，特征值问题（其中特征参数和多个有效本征模式之一必须同时求解）仍然具有挑战性，因为标准的算子学习框架假设存在唯一的输入-输出映射。本研究证明，傅里叶神经算子（FNO）结合基于小波的偏微分方程输入编码，能够学习并预测弹性波方程的多个本征模式，这些模式对应于声波在任意超材料几何结构中传播的形变模式。我们提供了机理性的解释和实验证据，说明为何小波编码与FNO的双重空间-频谱结构高度匹配，从而能够在单一模型中对连续值几何和二元值几何实现确定性的模式选择，以及为何预测精度随几何不连续性而变化。对于超材料设计，所提出的代理模型在消费级CPU上将设计循环中的仿真阶段加速了三个数量级（相比有限元分析），同时保持高保真度。这些结果还对基于频谱神经算子的其他多模式偏微分方程求解器的输入编码设计具有更广泛的启示。

英文摘要

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.

发表机构

• Duke University（杜克大学）
• California Institute of Technology（加州理工学院）

机构由 AI 辅助整理，请以论文原文为准。