Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver

2026-02-20

AI Summary

This work proposes Neural-HSS, a parameter-efficient neural network architecture grounded in the hierarchical semi-separable (HSS) matrix structure, to address the limitations of deep learning-based partial differential equation (PDE) solvers in low-data regimes and their high computational cost. By leveraging the low-rank structure inherent in the Green’s functions of elliptic PDEs, Neural-HSS achieves high-accuracy solutions under extremely scarce data conditions. Theoretical analysis reveals intrinsic connections between the HSS framework, Fourier neural operators, and convolutional layers, elucidating its enhanced data efficiency. Empirical results demonstrate that Neural-HSS substantially outperforms existing methods on a three-dimensional Poisson equation with two million grid points and generalizes effectively across diverse PDE applications in electromagnetics, fluid dynamics, and biology.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• PDE solver
• data efficiency
• low-data regime
• computational cost
• deep learning

Innovation

• Hierarchical Semi-Separable
• data-efficient
• neural PDE solver
• Green's function
• low-data regime