DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEs
Yingjie Tan, Quanming Yao, Yaqing Wang
TL;DR
DGNet leverages Green's function theory to create a graph-based neural PDE solver that significantly improves data efficiency and generalization, requiring fewer training trajectories and performing well on unseen source terms.
Contribution
The paper introduces DGNet, a novel discrete Green network that incorporates Green's function theory into a graph-based neural architecture for efficient PDE learning.
Findings
Achieves state-of-the-art accuracy with limited training data
Exhibits robust zero-shot generalization to unseen source terms
Reduces data requirements compared to existing neural PDE solvers
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.…
Peer Reviews
Decision·ICLR 2026 Poster
- Solid derivation of a discrete Green update tied to an implicit Crank–Nicolson step, with practical “factorize once, solve many” sparse solves; the operator depends only on mesh and \Delta t, enabling reuse during rollout. - Well-motivated hybrid operator that anchors learning with physics priors (Green–Gauss gradient, cotangent Laplacian) plus GNN corrections; clear architectural overview. - Broad empirical coverage with consistent SOTA on both log-MSE and relative $\ell_2$ across diverse r
- Some implementation details (e.g., GPU sparse LU and custom adjoint) are in the appendix; a brief complexity/throughput table in the main text would help readers assess practicality across datasets. - Uses standard message-passing blocks; novelty is concentrated in the Green discretization + operator split, not in GNN mechanics. - Although there is an ablation, the residual GNN’s incremental benefit and the sensitivity to factorization accuracy (e.g., fill-in thresholds, preconditioners) c
The paper addresses a longstanding limitation in neural PDE solvers—their inability to generalize to unseen source terms—by explicitly incorporating the Green’s function concept into a learnable, graph-based framework. This formulation is conceptually elegant: by separating the effect of the source term from the system dynamics, DGNet captures the response structure in a principled and interpretable manner. The combination of physics-based discretization and neural correction strikes a strong ba
Despite its strong results, the paper suffers from several clarity and interpretability issues that limit its accessibility. First, the presentation of the operator $\mathcal{L}$ and its components lacks precision. Although equations define $\mathcal{L}{\text{physics}}$ and $\mathcal{L}{\text{NN}}$, it remains unclear how $\mathcal{L}{\text{physics}}$ is numerically computed and integrated into the overall update rule—whether gradients and Laplacians are used merely as input features or as direc
1. The focus on the source term is interesting. Since the source/forcing term is used across a wide range of research and engineering applications, the work is of great importance to the community. 2. The method is simple and reasonable. The authors successfully leverage classical theory to develop an effective approach for elegantly handling the source term.
1. The applicable domain and limitations of the work are unclear. Since the method relies on the superposition and linear approximation of the spatial operator $\mathcal{L}_\boldsymbol{x}$, the reviewer considers it to work only for semilinear PDEs, although the paper says “PDE” with no specifications. The authors should clarify the assumptions and limitations of the work. 2. Section 3.4.1 is unclearly written. The paper uses $A_i$ and $d_i$ for volume and area, respectively, but this is outside
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Taxonomy
TopicsModel Reduction and Neural Networks · Machine Learning in Materials Science · Generative Adversarial Networks and Image Synthesis