cs.LGFeb 20, 2026

Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver

Authors: Pietro Sittoni, Emanuele Zangrando, Angelo A. Casulli, Nicola Guglielmi, Francesco Tudisco

Organizations: Gran Sasso Science Institute, L’Aquila, Italy · University of Edinburgh, UK

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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.

Figures & tables

Figure 1: Model overview. The lifting and projection layers can be implemented either as HSS layers or as full-rank layers. We illustrate, as an example, the weight matrix structure with two different hierarchical levels.
Figure 2: Train size vs relative test error for different models. The models are trained on a 1D Poisson equation.
Figure 3: Left : Train size vs relative test error for different models. The models are trained on a 3D Poisson equation. Right : Timing of forward and backward for the different models.
 Neural-HSSFNOResNetDeepONetGreen Learning
EquationParamsTest ErrParamsTest ErrParamsTest ErrParamsTest ErrParamsTest Err
Heat Eq.4545 K3×10−33\times 10^{-3}150150 K8×10−38\times 10^{-3}165165 K1×10−21\times 10^{-2}247247 K1×10−21\times 10^{-2}8383 K1×10−21\times 10^{-2}
Poisson Neumann BC3737 K2×10−42\times 10^{-4}102102 K3×10−33\times 10^{-3}165165 K4×10−14\times 10^{-1}247247 K9×10−39\times 10^{-3}8383 K3×10−33\times 10^{-3}
Burgers’ Eq.1.51.5 M0.120.121.51.5 M0.260.261.71.7 M0.570.571.71.7 M0.440.44--
Table 1: Comparison between Neural-HSS and baseline models. We report the number of learnable parameters for each model and the test error Eq. 3 .
 Neural-HSSFNOResNetDeepONetGreen LearningFactFormer
EquationParamsTest ErrParamsTest ErrParamsTest ErrParamsTest ErrParamsTest ErrParamsTest Err
Poisson Eq.37K7×10−87\times 10^{-8}132132 K7×10−67\times 10^{-6}165165 K3×10−43\times 10^{-4}280280 K2×10−22\times 10^{-2}8383 K5×10−55\times 10^{-5}3.93.9 M6×10−36\times 10^{-3}
Poisson Eq. (Mixed BC)3737 K1×10−41\times 10^{-4}132132 K6×10−36\times 10^{-3}165165 K8×10−28\times 10^{-2}280280 K4×10−14\times 10^{-1}8383 K1×10−21\times 10^{-2}3.9 M7×10−27\times 10^{-2}
Poisson Eq. (L-shape domain)3737 K2×10−22\times 10^{-2}132132 K2×10−12\times 10^{-1}165165 K7×10−17\times 10^{-1}280280 K8×10−18\times 10^{-1}8383 K5×10−15\times 10^{-1}3.9 M2×10−12\times 10^{-1}
Helmholtz equation3737 K6×10−36\times 10^{-3}132132 K5×10−25\times 10^{-2}165165 K1×1001\times 10^{0}280280 K1×1001\times 10^{0}8383 K1×1001\times 10^{0}3.9 M7×10−27\times 10^{-2}
Gray–Scott Eq. (Forward)329K0.2942.1M0.3311.99M0.2732.3M0.315--3.9M0.291
Table 2: Comparison between Neural-HSS and baseline models on 2D problems. We report the number of learnable parameters for each model and the test error Equation 2 . With NS Eq., we refer to the incompressible Navier-Stokes equation.
Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Train size vs relative test error for different models. The models are trained on a 3D Poisson equation
Figure 5: Ablation study on rank, outer rank, and levels for the Gray-Scott problem, where the color indicates the test error after training (rescaled from maximum to minimum observed, notice the scale in the colorbars).
Figure 6: Trajectory test loss varying β\beta
Figure 7: Timing of one forward pass, calculated on different datasets.
Figure 8: Timing of one forward+backward pass, calculated on two different datasets: Left : Heat equation. Right : Burgers’ equation. We refer to Appendix K for inference only timings.
Figure 9: Timing of one forward+backward pass, calculated on two different datasets: Left : 2D Poisson Equation Right : Gray-Scott and Navier-Stokes.We refer to Appendix K for inference only timings.
ConfigurationNeural-HSSFNOResNetDeepONetGreen Learning
Depth3366121244 1 1 1 The depth is the same for the Trunk network and the Branch network66
Embedding Dimension25625632324848192192128128
ActivationTrainable LeakyReLUGeLUGeLUGeLURational activation 2 2 2 ( Boullé et al., 2020 )
Levels33----
Rank22----
Modes-1212---
Appendix
Table 3: Models and training configuration for the Heat Equation and Poisson Equation with Neumann boundary conditions.
ConfigurationNeural-HSSFNOResNetDeepONet
Depth4466121244
Embedding Dimension102410246464150150512512
ActivationTrainable LeakyReLUGeLUGeLUGeLU
Levels33---
Rank3232---
Modes-3232--
Appendix
Table 4: Models and training configuration for the Burgers’ Equation .
ConfigurationNeural-HSSFNOResNetDeepONetGreen LearningFactFormer
Depth33444444 3 3 footnotemark: 36611
Embedding Dimension64641616646464646464384384
ActivationTrainable LeakyReLUGeLUGeLUGeLURational activation 4 4 footnotemark: 4GeLU
Levels22-----
Rank22-----
outer rank88-----
Appendix
Table 5: Models and training configuration for the 2D Poisson Equation and 2D Poisson Equation (L-shape domain) .
ConfigurationNeural-HSSFNOResNetDeepONetGreen Learning 
Depth33444444 5 5 footnotemark: 56611
Embedding Dimension64641616646464646464384384
ActivationTrainable LeakyReLUGeLUGeLUGeLURational activation 6 6 footnotemark: 6GeLU
Levels22-----
Rank22-----
outer rank88-----
Appendix
Table 6: Models and training configuration for the Helmholtz equation .
ConfigurationNeural-HSSFNOResNetDeepONetFactFormer
Depth3344444411
Embedding Dimension128×128128\times 12832326464128128384384
ActivationTrainable LeakyReLUGeLUGeLUGeLUGeLU
Levels22----
Rank88----
Outer Rank88----
Appendix
Table 7: Models and training configuration for the Incompressible Navier-Stokes and Gray-Scott .
ConfigurationNeural-HSSFNOResNetDeepONet
Depth11222244
Embedding Dimension128×128×128128\times 128\times 12814144848128128
ActivationTrainable LeakyReLUGeLUGeLUGeLU
Levels22---
Rank44---
Outer Rank22---
Appendix
Table 8: Models and training configuration for the data efficiency for 3D Poisson Equation . We report the training steps since for each training size we match the training steps at 1616 k.
ConfigurationNeural-HSSFNOResNetDeepONetGreen Learning
Depth3366121244 7 7 footnotemark: 766
Embedding Dimension25625632324848192192128128
ActivationTrainable LeakyReLUGeLUGeLUGeLURational activation 8 8 footnotemark: 8
Levels33----
Rank22----
Modes-1212---
Appendix
Table 9: Models and training configuration for the data efficiency for 1D Poisson Equation .

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