DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEs

Yingjie Tan; Quanming Yao; Yaqing Wang

cs.LGMar 2, 2026

DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEs

Authors: Yingjie Tan, Quanming Yao, Yaqing Wang

Organizations: Qiuzhen College, Tsinghua University · Department of Electronic Engineering, Tsinghua University · Beijing Institute of Mathematical Sciences and Applications

Abstract

Spatiotemporal partial differential equations (PDEs) underpin a wide range of scientific and engineering applications. Neural PDE solvers offer a promising alternative to classical numerical methods. However, existing approaches typically require large numbers of training trajectories, while high-fidelity PDE data are expensive to generate. Under limited data, their performance degrades substantially, highlighting their low data efficiency. A key reason is that PDE dynamics embody strong structural inductive biases that are not explicitly encoded in neural architectures, forcing models to learn fundamental physical structure from data. A particularly salient manifestation of this inefficiency is poor generalization to unseen source terms. In this work, we revisit Green's function theory-a cornerstone of PDE theory-as a principled source of structural inductive bias for PDE learning. Based on this insight, we propose DGNet, a discrete Green network for data-efficient learning of spatiotemporal PDEs. The key idea is to transform the Green's function into a graph-based discrete formulation, and embed the superposition principle into the hybrid physics-neural architecture, which reduces the burden of learning physical priors from data, thereby improving sample efficiency. Across diverse spatiotemporal PDE scenarios, DGNet consistently achieves state-of-the-art accuracy using only tens of training trajectories. Moreover, it exhibits robust zero-shot generalization to unseen source terms, serving as a stress test that highlights its data-efficient structural design.

Figures & tables

Figure 1: Overview of DGNet architecture. The model centers on a hybrid operator $\mathbf{L}=\mathbf{L}_{\text{physics}}+\mathbf{L}_{\text{neural}}$ , where $\mathbf{L}_{\text{physics}}$ encodes gradient and Laplacian discretizations and $\mathbf{L}_{\text{neural}}$ is a GNN-based correction for mesh-induced errors. This operator is integrated into the discrete Green’s function update (Eq. 7 ), which naturally combines system evolution with source-term response and provides structural inductive bias for data-efficient learning.

Figure 2: Geometric variables for the discrete operator.

Scenario	Metric	DeepONet	MGN	MP-PDE	BENO	PhyMPGN	DGNet
Allen- Cahn	MSE	2.60e-01	2.70e-01	8.52e-01	2.52e+00	5.16e-01	8.75e-03
Allen- Cahn	RNE	0.6686	0.6813	1.2109	2.0813	0.9420	0.0188
Fisher- KPP	MSE	3.05e-02	3.66e-03	9.90e-02	6.26e-02	1.50e-02	2.59e-04
Fisher- KPP	RNE	0.4181	0.1448	0.7530	0.5989	0.9270	0.0238
FitzHugh- Nagumo	MSE	2.49e-06	3.75e-05	6.464e-06	2.14e-04	1.69e-03	1.18e-07
FitzHugh- Nagumo	RNE	0.9745	3.4815	1.4454	8.3106	23.5696	0.0952

Table 2: Results across scenarios from three categories of spatiotemporal PDE systems under the limited-trajectory training. For both MSE (in log scale) and RNE, lower values indicate better performance. The best results are highlighted in bold.

Panel of Figure 3 — Figure 3: Visualization of prediction results on classical PDE scenarios. Rows from top to bottom correspond to the Allen–Cahn, FitzHugh–Nagumo, and Fisher–KPP equations, respectively.

Panel of Figure 4 — Figure 4: Visualization of prediction results on scenarios with complex geometric domains. Rows from top to bottom correspond to the cylinder, sediment, and complex obstacle cases, respectively.

Panel of Figure 5 — Figure 5: Visualization of generalization performance on the laser heat scenario with unseen sources.

Figure 6: Ablation study on the complex obstacle scenario.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Classical PDEs
Dataset	Domain	Nodes (approx.)	$\bm{\Delta t}$	Steps	Traj. (train/test)
Allen–Cahn	$[0,1]^{2}$	1,296	0.005	1,000	10 / 10
Fisher–KPP	$[0,1]^{2}$	1,681	0.00125	1,600	10 / 10
FitzHugh–Nagumo	$[0,1]^{2}$	10,000	0.005	1,200	7 / 3
Complex Geometric Domains
Cylinder	$[0,24]\times[0,8]$	2,275	0.05	2,000	15 / 5

Appendix

Table 3: Simulation parameters of all datasets used in experiments.

Scenario	Batch Size	Initial LR	Epochs	Decay Step	Decay Rate	Sub-seq. Length	Train/Test Trajs.
Allen–Cahn	36	1e-3	600	200	0.1	20	10/10
Fisher–KPP	24	5e-4	300	60	0.1	10	10/10
FitzHugh–Nagumo	10	1e-3	400	100	0.1	10	7/3
Cylinder	6	5e-4	300	100	0.2	6	15/5
Sediments	6	5e-4	300	100	0.2	6	15/5
Complex Obstacles	6	5e-4	300	100	0.2	6	15/5

Appendix

Table 4: Training hyperparameters for each experimental scenario.

Figure 7: Comparison of the learned operator with the reference. Left: input physical field. Center: reference operator computed by finite difference. Right: operator predicted by DGNet.

Figure 8: Heatmap visualization of operator matrices. Left: $\mathbf{L}_{\text{physics}}$ . Middle: $\mathbf{L}_{\text{neural}}$ . Right: final corrected operator $\mathbf{L}$ .

Figure 9: Scalability of inference time and peak memory with respect to mesh size.

Dataset	DeepONet	MGN	MP-PDE	BENO	PhyMPGN	DGNet
Allen-Cahn	1.33	5.12	5.96	14.06	6.71	8.37
Fisher-KPP	2.77	12.74	19.20	75.92	17.40	15.76
FitzHugh-Nagumo	106.52	15.48	45.23	67.32	10.72	23.78
Cylinder	6.38	20.15	24.90	133.01	12.47	18.10
Sediments	58.94	21.47	31.25	101.10	14.59	21.63
Complex Obstacles	141.92	23.59	47.36	108.96	20.31	29.08

Appendix

Table 6: Comparison of single-trajectory inference time (in seconds) between DGNet and baseline methods.

Figure 10: Ablation study results on the FitzHugh–Nagumo scenario.

Figure 11: Ablation study results on the Laser Heat scenario.

Figure 12: Eigenvalue spectrum analysis for the FitzHugh–Nagumo task. Blue indicates the physics prior $\mathbf{L}_{physics}$ , and orange indicates the total hybrid operator $\mathbf{L}$ .

Figure 13: Eigenvalue spectrum analysis for the Laser Heat task. Blue indicates the physics prior $\mathbf{L}_{physics}$ , and orange indicates the total hybrid operator $\mathbf{L}$ .

Figure 14: Speed-accuracy comparison between DGNet and the classical FEM numerical solver on the FitzHugh-Nagumo task.

Figure 15: Visualization of the 3D heat conduction scenario, showing the mechanical bracket geometry.

Metric	DeepONet	MGN	MP-PDE	BENO	PhyMPGN	DGNet
MSE	4.41e+03	4.60e+02	2.59e+04	5.28e+02	8.83e+03	4.42e+01
RNE	0.1630	0.1249	0.1993	0.0938	0.2352	0.0314

Appendix

Table 7: Quantitative results on the 3D laser-heat task. DGNet achieves the lowest MSE and RNE compared to baselines.

Figure 16: Visualization of the modified wavy channel scenario ( $Re=300$ ).

Metric	DeepONet	MGN	MP-PDE	BENO	PhyMPGN	DGNet
MSE	2.38e-01	3.59e-03	7.94e-03	2.30e-02	7.95e-02	6.29e-04
RNE	0.8539	0.3591	0.4729	0.6292	1.1838	0.0385

Appendix

Table 8: Quantitative comparison on the high-Reynolds (

Re=300

) fluid and pollutant transport task.

Figure 17: Snapshot of the light-driven reaction scenario.

Metric	DeepONet	MGN	MP-PDE	BENO	PhyMPGN	DGNet
MSE	6.11e+03	1.93e+02	1.37e+01	6.04e+02	2.87e+02	3.97e+00
RNE	0.8832	0.1582	0.0422	0.2796	0.1921	0.0227

Appendix

Table 9: Quantitative results on the large-scale light-driven reaction task.

Source Attribute	0.5 $\times$	1 $\times$	2 $\times$	3 $\times$
bandwidth	1.53e+01	1.76e+01	3.88e+01	7.36e+01
amplitude	2.27e+01	1.76e+01	1.62e+01	1.95e+01
number of concurrent sources	1.09e+01	1.76e+01	3.26e+01	4.40e+01

Appendix

Table 10: Robustness to source characteristics. Each column scales the corresponding attribute by the indicated factor relative to training.

Figure 18: Visualization of the failure mode in the quasilinear Phase Transition scenario.

Dataset	Metric	SINGER	AMG (Harnessing)	Transolver	DGNet
FitzHugh-Nagumo	MSE	4.58e-06	3.30e-05	7.87e-06	1.18e-07
FitzHugh-Nagumo	RNE	1.3844	2.4837	0.6729	0.0952
Complex Obstacles	MSE	9.41e-03	1.40e-03	8.94e-04	6.69e-05
Complex Obstacles	RNE	0.7746	0.3058	0.1855	0.0211
Laser Heat	MSE	9.58e+03	2.96e+03	5.72e+03	1.76e+01
Laser Heat	RNE	0.2504	0.1831	0.1785	0.0102

Appendix

Table 11: Quantitative comparison with recent state-of-the-art baselines (SINGER, AMG-GNN, Transolver) on representative tasks.

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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.

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