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
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 L=Lphysics+Lneural , where Lphysics encodes gradient and Laplacian discretizations and Lneural 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
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
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
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.
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.
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.
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
Dataset
Domain
Nodes (approx.)
Δt
Steps
Traj. (train/test)
Classical PDEs
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]×[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: Lphysics . Middle: Lneural . Right: final corrected operator 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 Lphysics , and orange indicates the total hybrid operator L .
Figure 13: Eigenvalue spectrum analysis for the Laser Heat task. Blue indicates the physics prior Lphysics , and orange indicates the total hybrid operator 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 ×
1 ×
2 ×
3 ×
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
RNE
1.3844
2.4837
0.6729
0.0952
Complex Obstacles
MSE
9.41e-03
1.40e-03
8.94e-04
6.69e-05
RNE
0.7746
0.3058
0.1855
0.0211
Laser Heat
MSE
9.58e+03
2.96e+03
5.72e+03
1.76e+01
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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