Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver

Pietro Sittoni, Emanuele Zangrando, Angelo Alberto Casulli, Nicola Guglielmi, Francesco Tudisco
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:113805-113831, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-sittoni26a, title = {Neural-{HSS}: Hierarchical Semi-Separable Neural {PDE} Solver}, author = {Sittoni, Pietro and Zangrando, Emanuele and Casulli, Angelo Alberto and Guglielmi, Nicola and Tudisco, Francesco}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {113805--113831}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/sittoni26a/sittoni26a.pdf}, url = {https://proceedings.mlr.press/v306/sittoni26a.html}, 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.} }
Endnote
%0 Conference Paper %T Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver %A Pietro Sittoni %A Emanuele Zangrando %A Angelo Alberto Casulli %A Nicola Guglielmi %A Francesco Tudisco %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-sittoni26a %I PMLR %P 113805--113831 %U https://proceedings.mlr.press/v306/sittoni26a.html %V 306 %X 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.
APA
Sittoni, P., Zangrando, E., Casulli, A.A., Guglielmi, N. & Tudisco, F.. (2026). Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:113805-113831 Available from https://proceedings.mlr.press/v306/sittoni26a.html.

Related Material