Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver
Pietro Sittoni, Emanuele Zangrando, Angelo A. Casulli, Nicola Guglielmi, Francesco Tudisco
TL;DR
Neural-HSS is a data-efficient neural PDE solver leveraging Hierarchical Semi-Separable matrices, achieving high accuracy with limited data across various PDE applications.
Contribution
The paper introduces Neural-HSS, a novel architecture that is provably data-efficient and structurally inspired by Green's functions, with theoretical analysis and broad PDE applicability.
Findings
Outperforms baseline methods in low-data regimes
Successfully solves 3D Poisson equation with 2 million points
Learns from diverse PDE data in electromagnetism, fluid dynamics, biology
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…
Peer Reviews
Decision·ICLR 2026 Conference Desk Rejected Submission
The single block is algebraically identical to one Schulz step, i.e., $(2I-A_0A)A_0b$, grounding the design in a classical fixed-point map rather than heuristics. The residual objective avoids needing $A^{-1}$; depth clearly helps on hard spectra: for a discrete Helmholtz matrix, $k=3$ reaches $\epsilon_{\text{inv}}\approx1.16\times10^{-4}$ while $k\le2$ underperforms. The perturbation protocol—$A+\varepsilon I$ and $A+R$ with $r_{ij}\sim U(-\varepsilon,\varepsilon)$—demonstrates adaptation down
The method is trained per matrix $A$, and the paper does not examine whether a model generalizes across a family of matrices. There are no guarantees for the trained deep composition, and the threshold between successful and failed fine-tuning is only reported empirically. When used as a preconditioner, application-level comparisons in terms of wall-clock time and iteration counts against strong baselines are not provided. Efficiency claims rely on IDBF-type hierarchical low-rank stru
Theorems 2.2 and 2.3 provide the solid theoretical foundation, proving that convolutional kernel can be approximated by HSS and exact recovery and data-efficiency. The experiments are comprehensive testing several types of PDEs and compare with the SOTA methods.
It seems that the method only support Dirichlet boundary condition. Can it support non-zero Dirichlet boundary conditions? The method only supports elliptic PDEs. The Green's functions of other PDEs are not semi low rank, so it is hard to be extended to other PDEs.
- **Theoretical novelty**: The idea of learning HSS-style hierarchical structures within NNs is insightful. - **Interpretable architecture**: Each network component has a clear algebraic analog.
- **Restricted system matrix formulations**: All tests use symmetric positive-definite or weakly oscillatory kernels. Whether the proposed methods work on highly indefinite or ill-conditioned systems (e.g., high-frequency Helmholtz problems) is unclear. - **Insufficient baselines**: Comparisons with legacy solvers, e.g., legacy GMERS (with or without learned preconditioners), NVIDIA AmgX, etc., are missing. Without a significant performance boost over legacy routines, it remains questionable why
The paper is strong in both conceptual innovation and methodological rigor. It introduces a well-motivated and theoretically grounded neural operator architecture that effectively integrates structural priors from numerical analysis into deep learning. The work stands out for its clear theoretical foundations, demonstrating provable guarantees, and for its strong empirical performance, showing consistent efficiency and scalability across diverse PDE tasks.
The paper lacks detailed ablation or interpretability studies to elucidate how specific HSS components contribute to overall performance, leaving uncertainty about which architectural factors are most critical. Moreover, the experimental comparison focuses primarily on FNO, ResNet, and DeepONet, without including more recent neural operator baselines, which somewhat limits the empirical breadth and generality of the conclusions.
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Taxonomy
TopicsModel Reduction and Neural Networks · Generative Adversarial Networks and Image Synthesis · Machine Learning in Materials Science