mlsanityData to 7 Oct 2026

Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver

arXiv 2602.18248 ↗First posted 20 Feb 2026cs.LG cs.NA math.NAcited by 1

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Abstract (arXiv)

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.

Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 40 of its 71 references that are in the map. First posted on arXiv 20 Feb 2026.

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40 of 71 references.

References · 40

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2021Swin Transformer: Hierarchical Vision Transformer using Shifted Windowscited by 12,082
  3. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  5. 2022Message Passing Neural PDE Solverscited by 368
  6. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 321
  7. 2014Speeding-up Convolutional Neural Networks Using Fine-tuned CP-Decompositioncited by 199
  8. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  9. 2024The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learningcited by 134
  10. 2019Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networkscited by 127
  11. 2017Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systemscited by 125
  12. 2023Scalable Transformer for PDE Surrogate Modelingcited by 123
28 more
  1. 2020Transformers for Modeling Physical Systemscited by 105
  2. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 100
  3. 2018Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flowcited by 97
  4. 2022Predicting Physics in Mesh-reduced Space with Temporal Attentioncited by 82
  5. 2019Learning to Optimize Multigrid PDE Solverscited by 79
  6. 2020Rational neural networkscited by 75
  7. 2023A Mathematical Guide to Operator Learningcited by 74
  8. 2019Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizationscited by 73
  9. 2023Monarch Mixer: A Simple Sub-Quadratic GEMM-Based Architecturecited by 62
  10. 2021Pufferfish: Communication-efficient Models At No Extra Costcited by 61
  11. 2022CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representationscited by 56
  12. 2024On conditional diffusion models for PDE simulationscited by 56
  13. 2020Learning Algebraic Multigrid Using Graph Neural Networkscited by 54
  14. 2023Towards Stability of Autoregressive Neural Operatorscited by 54
  15. 2022Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equationscited by 42
  16. 2021Learning Dissipative Dynamics in Chaotic Systemscited by 41
  17. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 33
  18. 2025PDE-Transformer: Efficient and Versatile Transformers for Physics Simulationscited by 29
  19. 2025Generative Latent Neural PDE Solver using Flow Matchingcited by 28
  20. 2023Elliptic PDE learning is provably data-efficientcited by 26
  21. 2021Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learningcited by 21
  22. 2018Learning Compressed Transforms with Low Displacement Rankcited by 19
  23. 2025Appa: Bending Weather Dynamics with Latent Diffusion Models for Global Data Assimilationcited by 17
  24. 2022A Deep Conjugate Direction Method for Iteratively Solving Linear Systemscited by 12
  25. 2023Geometry-aware training of factorized layers in tensor Tucker formatcited by 12
  26. 2017Theoretical Properties for Neural Networks with Weight Matrices of Low Displacement Rankcited by 11
  27. 2021Accelerating GMRES with Deep Learning in Real-Timecited by 7
  28. 2024Fast Tree-Field Integrators: From Low Displacement Rank to Topological Transformerscited by 3

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Citing papers per year: 2026: 1.

1 paper.

  1. 2026Transferability of Learned States in Neural PDE Solverscited by 0