New Graph-Based Spectral Loss Predicts Chaos on Unstructured Meshes

Surrogate modeling of chaotic, high-dimensional dynamical systems traditionally relies on structured grids where Fourier modes enable clean spectral breakdowns. But real-world simulations—from climate to aerodynamics—often use irregular meshes that lack a canonical Fourier basis. In this paper, Kanad Sen and Romit Maulik introduce a scale-aware learning framework that replaces Fourier-band power losses with graph-Laplacian frequency bands. By constructing spectral representations from mesh connectivity and geometry, the method preserves both pointwise accuracy and the scale-dependent structure of physical fields, a critical requirement for chaos-prone systems.

To keep the approach scalable, the authors propose two approximations: Chebyshev polynomial graph filters that avoid explicit eigendecomposition, and GLEAM (Graph Laplacian Energy Alignment for Meshes), which applies retained-subspace supervision across multilevel graph hierarchies. This ensures coarse and fine representations are regularized during autoregressive rollout. Experiments on forecasting turbulent flows on unstructured meshes show that the proposed spectral losses improve long-horizon fidelity and preserve statistical invariants, outperforming deterministic baselines. The work bridges graph neural networks and physics-informed ML, enabling accurate AI predictions for complex, real-world geometries.