Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver
This addresses the problem of high computational costs in PDE simulations for applications like electromagnetism and fluid dynamics, though it is incremental as it builds on existing HSS matrix structures.
The paper tackles the computational cost of generating large-scale datasets for training deep learning-based PDE solvers by introducing Neural-HSS, a parameter-efficient architecture that is provably data-efficient for elliptic PDEs, demonstrating superior learning from low data on a 3D Poisson equation with two million points and outperforming baselines.
Deep learning-based methods have shown remarkable effectiveness in solving PDEs, largely due to their ability to enable fast simulations once trained. However, despite the availability of high-performance computing infrastructure, many critical applications remain constrained by the substantial computational costs associated with generating large-scale, high-quality datasets and training models. In this work, inspired by studies on the structure of Green's functions for elliptic PDEs, we introduce Neural-HSS, a parameter-efficient architecture built upon the Hierarchical Semi-Separable (HSS) matrix structure that is provably data-efficient for a broad class of PDEs. We theoretically analyze the proposed architecture, proving that it satisfies exactness properties even in very low-data regimes. We also investigate its connections with other architectural primitives, such as the Fourier neural operator layer and convolutional layers. We experimentally validate the data efficiency of Neural-HSS on the three-dimensional Poisson equation over a grid of two million points, demonstrating its superior ability to learn from data generated by elliptic PDEs in the low-data regime while outperforming baseline methods. Finally, we demonstrate its capability to learn from data arising from a broad class of PDEs in diverse domains, including electromagnetism, fluid dynamics, and biology.