Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver

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.

Modern science relies heavily on computer simulations to understand the physical world, from predicting how heat spreads through a material to modeling fluid flow. These simulations are accurate but slow and expensive to run. Machine learning offers a possible alternative: a model is trained on a set of simulations and can then produce new predictions much more quickly. A major challenge is that both generating simulation data and training such models can require large amounts of computation, especially for three-dimensional problems. Many physical systems have a useful mathematical structure. Points that are close to each other in space interact in detailed ways, while interactions between distant regions can often be represented in a compressed form with far fewer numbers. Neural-HSS is a neural network designed around this idea. It allocates most of its capacity to local interactions, while representing long-range interactions in a compressed way. This makes it more efficient to train and reduces the amount of data needed. Because the model matches the structure of the underlying physics, it can reach high accuracy using significantly fewer training simulations, up to eight times fewer than other recent methods, while also training faster. It also performs well across a range of problems, including fluid dynamics, electromagnetism, and biological systems.