Hypergraph Wavelet Operators Advance PDE Solution for Complex Geometries

Traditional methods for modeling physical systems, especially those involving Partial Differential Equations (PDEs), often struggle with complex geometries and unstructured data, where grid-centric representations lose accuracy. While graph neural operators address some of these challenges, their reliance on pairwise edges fails to capture more intricate group-wise interactions within mesh cells or conservation volumes. This new research presents the Hypergraph Adaptive Wavelet Operator (HALO), an innovative approach that elevates the problem domain to a hypergraph structure. HALO learns within this hypergraph's spectral wavelet domain, avoiding explicit hypergraph-Laplacian eigendecomposition by using Chebyshev polynomial wavelet filters. This design provides localized spectral kernels at a linear sparse-matrix cost. HALO's trainable dyadic wavelet scales are regularized to ensure tight-frame coverage, allowing the frequency response to adapt to specific PDEs while maintaining stable multi-scale spectral coverage. Across various 2D and 3D benchmarks, including industrial aerodynamic geometries with hundreds of thousands of points, HALO consistently achieves leading or near-leading accuracy compared to other advanced neural operators, demonstrating stable multi-step rollouts and resolution-equivariant performance.