aiqbench analysis
This research advances surrogate modeling for chaotic systems by adapting spectral loss functions from structured grids to unstructured meshes using graph operators and Chebyshev polynomial filters. The proposed method, GLEAM, improves long-horizon forecasting fidelity for turbulent flows while preserving statistical invariants without the computational cost of full eigendecomposition.
Scale-Aware Learning of Chaotic Dynamics on Unstructured Meshes via Binned Spectral Losses
arXiv:2607.19387v1 Announce Type: cross Abstract: 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.
Abstract: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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computational Physics (physics.comp-ph); Fluid Dynamics (physics.flu-dyn)
arXiv:2607.19387 [cs.LG]
(or arXiv:2607.19387v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.19387 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 329 words
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