Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver

Neural PDE solvers can make simulation inexpensive after training, but producing high-quality training data and fitting large models remain costly. Neural-HSS addresses this bottleneck with an architecture based on hierarchical semi-separable matrix structure, motivated by the structure of Green’s functions for elliptic PDEs.

The authors establish exactness properties for the architecture in very low-data regimes and relate its operators to Fourier neural operator and convolutional layers. On a three-dimensional Poisson problem with a two-million-point grid, Neural-HSS learns effectively from limited data and outperforms the evaluated baselines in the low-data regime.

Experiments spanning electromagnetism, fluid dynamics, and biology indicate that the same structured approach applies across a broad range of PDE-driven systems.

Paper: https://arxiv.org/abs/2602.18248