Organizations: Faculty of Digital Innovation, Arts & Sciences, Saskatchewan Polytechnic, Regina SK S4S 5X1, Canada · Civil and Environmental Engineering, University of Alberta, Edmonton AB T6G 2H5, Canada · Energy Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, SK S4S 0A2, Canada · Industrial Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, SK S4S 0A2, Canada
Green's function based operator approximation offers an efficient route for solving linear partial differential equations under varying boundary conditions and source terms. Once the Green's function is learned, solutions for new configurations are obtained through integration rather than by solving the differential equation again. Existing Green's function learning methods face two structural challenges. The first is the singular behavior of the Green's function near the source point, which places a difficult approximation burden on neural networks. The second is the absence of explicit consistency between the learned Green's function and its gradient, although both quantities enter the integral solution representation directly. This work proposes SDC-GON, a Singular Decomposition and Consistency-Regularized Green's Operator Network that addresses both challenges within a unified framework. The Green's function is decomposed into an analytically known singular component and a smooth correction learned by the network, so that the neural approximation targets only the regular part of the response kernel. A self-consistency loss enforces agreement between the gradient and the autodifferentiation gradient of the smooth correction. The method is evaluated on two dimensional Poisson, three dimensional heat conduction, heterogeneous reaction diffusion, and Stokes benchmarks, consistently outperforming the compared baselines across all cases. On the heterogeneous pipe benchmark, SDC-GON achieves a testing error of 3.70×10−4 with a smaller network architecture, compared with 9.60×10−4 for the same-width baseline and 4.63×10−4 for a larger configuration, demonstrating that structural improvements are more effective than increasing model size.
Figure 2: Comparison of the original GON architecture and the proposed SDC-GON architecture.
Method
Backbone
Loss
Integration
BC
Source
Dim.
Singularity
Consistency
Boulle et al. [ 4 ]
Rational NN
Data
Monte Carlo
No
Yes
1D–2D
Indirect
No
Teng et al. [ 40 ]
FNN
PDE
Gauss
Yes
Yes
1D–2D
Indirect
No
Aldirany et al. [ 1 ]
FNN
PDE
Monte Carlo
No
Yes
1D–2D
Indirect
No
GON [ 14 ]
BsNN
Data
Gauss
Yes
Yes
3D
Indirect
No
SDC-GON
Shared BsNN
Data + consistency
Gauss
Yes
Yes
3D
Analytical
Yes
Table 1: Methodological comparison of representative Green’s function learning methods and SDC-GON. In the Singularity column, “Indirect” indicates that no known singular component is analytically subtracted; instead the singularity is handled implicitly through architectural flexibility (rational NN, BsNN) or smooth-surrogate training. “Analytical” indicates explicit subtraction of the free-space singular kernel before neural approximation.
Method
Training error
Testing error
Boulle et al. [ 4 ]
2.51×10−3
2.56×10−3
Teng et al. [ 40 ]
2.63×10−2
1.05×10−1
Aldirany et al. [ 1 ]
9.75×10−3
3.91×10−2
GON [ 14 ]
2.32×10−5
3.14×10−5
SDC-GON
4.86×10−8
8.64×10−8
Table 2: Comparison with existing Green’s function based methods for the two dimensional Poisson case.
Figure 3: Two dimensional Poisson case: absolute pointwise error for SDC-GON (a) and GON (b). Both panels use the same error color scale, and the circled regions indicate areas where the visual difference between the two methods is most evident.
Model
Layers
Training error
Testing error
PINN [ 28 ]
[3,12,12,12,1]
not reported
2.56×10−2
PI-DeepONet [ 41 ]
Trunk [3,12,12,12,1]
4.14×10−4
1.78×10−3
DeepONet [ 21 ]
Trunk [3,12,12,12,1]
8.14×10−4
1.37×10−3
FNO [ 19 ]
[2,3,3,3,3,3,1]
1.01×10−4
6.32×10−4
GON [ 14 ]
[6,12,12,12,1]
1.50×10−5
5.00×10−5
SDC-GON
[6,12,12,12,1]
7.30×10−6
8.20×10−6
Table 3: Performance of different baseline models on the three dimensional heat conduction case on the finned tube geometry.
Figure 4: Three dimensional heat conduction case: exact solution field (a), pointwise error for SDC-GON (b), and pointwise error for GON (c) on the finned tube geometry. Highlighted regions indicate areas of visible error reduction.
Figure 5: Computational meshes for the flat plate geometry (a) and the pipe geometry (b). Both reaction diffusion cases use a Gaussian random field source term f(x)=G(x)W/m3 .
Model
Layers
Training error
Testing error
PINN [ 28 ]
[3,12,12,12,1]
not reported
6.54×10−2
PI-DeepONet [ 41 ]
Trunk [3,12,12,12,1]
4.02×10−2
3.93×10−2
DeepONet [ 21 ]
Trunk [3,12,12,12,1]
3.68×10−2
3.45×10−2
FNO [ 19 ]
[2,6,6,6,6,1]
1.33×10−3
2.89×10−3
GON [ 14 ]
[6,12,12,12,1]
3.80×10−4
2.52×10−3
SDC-GON
[6,12,12,12,1]
1.00×10−4
2.78×10−4
Table 4: Performance of different baseline models on the flat plate reaction diffusion case.
Figure 6: Flat plate case: exact solution field (a), pointwise error for SDC-GON (b), and pointwise error for GON (c). Highlighted regions show localized error differences between the two methods.
Model
Layers
Training error
Testing error
PINN [ 28 ]
[3,24,24,24,1]
not reported
1.64
PI-DeepONet [ 41 ]
Trunk [3,24,24,24,1]
4.72×10−3
4.63×10−2
DeepONet [ 21 ]
Trunk [3,24,24,24,1]
1.95×10−3
2.84×10−2
FNO [ 19 ]
[2,12,12,12,12,1]
1.72×10−3
2.14×10−3
GON small [ 14 ]
[6,12,12,12,1]
2.20×10−4
9.60×10−4
GON large [ 14 ]
[6,24,24,24,1]
4.43×10−4
4.63×10−4
Table 5: Performance of different baseline models on the pipe reaction diffusion case.
Figure 7: Pipe case: exact solution field (a), pointwise error for SDC-GON (b), and pointwise error for GON (c). SDC-GON uses the smaller BsNN architecture, while GON is shown for comparison using the same error scale.
Model
Layers
Training error
Testing error
PINN [ 28 ]
[3,12,12,12,1]
not reported
1.08×10−2
PI-DeepONet [ 41 ]
Trunk [3,12,24,12,4]
5.59×10−3
5.31×10−3
DeepONet [ 21 ]
Trunk [3,12,24,12,4]
2.15×10−3
2.17×10−3
FNO [ 19 ]
[2,12,12,12,12,3]
1.36×10−3
2.21×10−3
GON small [ 14 ]
[6,4,6,8,1]
2.15×10−3
1.97×10−3
GON large [ 14 ]
[6,12,24,12,1]
5.71×10−4
5.83×10−4
Table 6: Performance of different baseline models on the Stokes cavity case.
Singular network
Singular decomposition
LR decay
Consistency loss
Training error
Testing error
✗
✗
✗
✗
3.80×10−4
2.52×10−3
✗
✗
✗
✓
4.60×10−4
7.40×10−4
✗
✗
✓
✗
4.80×10−4
6.30×10−4
✗
✓
✗
✗
3.70×10−4
6.90×10−4
✓
✗
✗
✗
3.70×10−3
1.10×10−3
✓
✓
✗
✗
4.80×10−4
5.60×10−4
Table 7: Ablation study on the reaction diffusion benchmark.
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.
Yingjie Tan, Quanming Yao, Yaqing Wang
Qiuzhen College, Tsinghua University · Department of Electronic Engineering, Tsinghua University · Beijing Institute of Mathematical Sciences and Applications
Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels using the Universal Spectral Basis (USB), a fixed, global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. Motivated by our theoretical finding that the discrete Green's functions of shift-invariant PDE discretizations exhibit spatial Linear Dynamical System (LDS) structure, we prove that these kernels admit compact approximations in the USB. By learning only the spectral coefficients of rapidly decaying eigenvalues, SFO achieves a highly efficient representation. Across six benchmarks, including reaction-diffusion, fluid dynamics, and 3D electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters.
Noam Koren, Rafael Moschopoulos, Kira Radinsky +1
Department of Computer Science Technion - Israel Institute of Technology · Department of Computer Science Princeton University
Learning solution operators for partial differential equations (PDEs) on irregular and geometry-dependent domains remains a central challenge in scientific machine learning. While spectral methods provide strong inductive biases for modeling global interactions, they are typically limited to regular domains, and existing neural approaches often require domain warping, interpolation, or costly geometric embeddings. We introduce the Graph Spectral Neural Operator (GSNO), a neural operator that combines spatial graph spectral decompositions with temporal Fourier transforms through a unified space--time spectral kernel. This formulation enables globally coherent operator learning on non-Cartesian discretizations without domain warping or autoregressive rollouts. By replacing learned geometric embeddings with a graph Laplacian spectral basis, GSNO provides geometry-aware spectral learning with low parameter complexity. Across steady and unsteady PDE benchmarks on irregular and geometry-dependent domains, GSNO achieves strong accuracy with reduced runtime and parameter counts, while demonstrating robust zero-shot generalization across mesh resolutions and geometry families.
Abdolmehdi Behroozi, Chaopeng Shen
Department of Civil and Environmental Engineering, Penn State University, University Park, PA 16802, USA