Beyond Pairwise Graphs in Science: Hypergraph Adaptive Wavelet Operators for Parametric PDEs

Researchers introduced the Hypergraph Adaptive waveLet Operator (HALO), a neural operator that learns solution maps for parametric PDEs by lifting the domain to a hypergraph and operating in its spectral wavelet domain, avoiding explicit eigendecomposition via Chebyshev polynomial filters. On 2D and 3D benchmarks, HALO achieved best or near-best accuracy among frequency-, transformer-, DeepONet-, state-space-, and graph-based baselines, and scaled to industrial aerodynamic meshes of a few hundred thousand points, matching or exceeding fixed-discretization transformers while remaining resolution-equivariant.

arXiv:2608.27883v1 Announce Type: new Abstract: Physical systems are often modeled by solution operators that map input fields, parameters, geometries, or past states to steady or future physical states. Learning these maps is difficult, especially for time-dependent systems that must assimilate history and remain stable under autoregressive rollout. Many neural operators work best on regular, structured grids, while realistic simulations often require unstructured meshes or point clouds to resolve complex geometries; in such settings, grid-centric representations can lose accuracy. Graph neural operators handle these domains through message passing or spectral graph filtering, but pairwise edges do not directly capture group-wise couplings among mesh cells, local neighborhoods, or conservation volumes. We introduce the Hypergraph Adaptive waveLet Operator HALO , which lifts the domain to a hypergraph and learns in its spectral wavelet domain. HALO avoids explicit hypergraph-Laplacian eigendecomposition through Chebyshev polynomial wavelet filters, giving localized spectral kernels at linear sparse-matrix cost. Its trainable dyadic wavelet scales are regularized toward tight-frame coverage, allowing the frequency response to adapt to each PDE while encouraging stable multi-scale spectral coverage. Across 2D and 3D benchmarks on structured and unstructured discretizations, HALO achieves best or near-best accuracy among frequency-, transformer-, DeepONet-, state-space-, and graph-based baselines and sustains stable multi-step rollouts. The same model scales to industrial aerodynamic geometries: on meshes of a few hundred thousand points it is on par with, or better than, the strongest fixed-discretization transformers, while remaining resolution-equivariant.