通过分箱谱损失在非结构化网格上进行混沌动力学的尺度感知学习

中文摘要 AI 辅助

高维非线性混沌动力系统的代理建模不仅需要保持逐点精度，还需保留物理场的尺度相关结构。带域谱功率损失（如分箱谱损失函数）在结构化网格上能提供这种监督，但非结构化网格没有规范的傅里叶基，需从网格连通性和几何诱导的图算子构建谱表示。本研究将分箱谱功率损失扩展到非结构化网格上的非线性动力系统代理建模，用图拉普拉斯频带取代傅里叶频带，并提供可扩展的切比雪夫和多级近似以提高长期预测保真度。全谱形式使用图拉普拉斯特征空间提供傅里叶带功率匹配的图类似物，但成本高，可扩展近似用稀疏切比雪夫多项式图滤波器取代精确带投影仪，避免显式特征分解。利用多级图架构时，引入网格图拉普拉斯能量对齐（GLEAM），跨图层次应用保留子空间尺度感知监督，使粗粒度和细粒度表示在自回归预测中得到正则化。结果表明，与确定性基线相比，所提出的谱损失提高了非结构化网格上湍流预测的长期预测保真度并保留了统计不变量。

英文摘要

Surrogate modeling for high-dimensional nonlinear dynamical systems that exhibit chaos requires mechanisms that preserve not only pointwise accuracy but also the scale-dependent structure of physical fields. Bandwise spectral power losses, such as the binned spectral loss function, provide such supervision on structured grids, where Fourier modes define a standard frequency decomposition. On irregular meshes, however, no canonical Fourier basis exists, and spectral representations must be constructed from graph operators induced by mesh connectivity and geometry. In this study, we extend the binned spectral power loss for application to unstructured-mesh surrogate modeling of nonlinear dynamical systems. This is obtained by replacing Fourier bands with graph-Laplacian frequency bands, and we provide scalable Chebyshev and multilevel approximations for improving long-horizon rollout fidelity. In its full-spectrum form, our approach uses graph Laplacian eigenspaces to provide a graph analogue of Fourier band-power matching, but incurs the high cost of spectral decomposition. As a scalable approximation, we replace exact band projectors with sparse Chebyshev polynomial graph filters, avoiding explicit eigendecomposition. When utilizing multilevel graph architectures, we introduce Graph Laplacian Energy Alignment for Meshes (GLEAM), which applies retained-subspace scale-aware supervision across graph hierarchies so that coarse and fine representations are regularized during autoregressive rollout. Our results show that the proposed spectral losses improve long-horizon rollout fidelity and preserve statistical invariants for the forecasting of turbulent flows on unstructured meshes, compared to deterministic baselines.

发表机构

• School of Mechanical Engineering, Purdue University（普渡大学机械工程学院）

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