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arXiv · 2602.18248

Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver

Pietro Sittoni·Emanuele Zangrando·Angelo A. Casulli·Nicola Guglielmi·Francesco Tudisco
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Abstract

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.

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Pietro Sittoni, Emanuele Zangrando, Angelo A. Casulli, Nicola Guglielmi, Francesco Tudisco. 2026-02-20. Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver. https://arxiv.org/abs/2602.18248

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