Beyond Pairwise Graphs in Science: Hypergraph Adaptive Wavelet Operators for Parametric PDEs
Machine Learning
Summary
The authors developed a new method called HALO to better predict how physical systems evolve over time, especially when dealing with complex shapes represented by unstructured meshes. Unlike other methods that look at pairs of points, HALO uses hypergraphs to capture group relationships and works in a special wavelet-based space to efficiently process data. It avoids costly calculations by using tricks called Chebyshev polynomial filters and adapts its focus on different scales to match each specific problem. In tests, HALO matched or outperformed other advanced techniques and handled large, realistic datasets while staying stable over multiple prediction steps.
Neural operatorsUnstructured meshesHypergraphsWaveletsChebyshev polynomialsSpectral filteringPDE (Partial Differential Equations)Multi-scale analysisAutoregressive rolloutsResolution equivariance
Authors
Rajat Sarkar, Venkataramana Runkana, Souvik Chakraborty
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.