Green's function networks improve solving partial differential equations
SDC-GON: Singular Decomposition and Consistency-Regularized Green's Operator Networks for Solving Partial Differential Equations
Machine Learning
Summary
Solving partial differential equations (PDEs) is important for modeling things like heat or fluid flow, but it can be slow when conditions change. The paper presents a method that learns a special function called Green's function, which helps solve PDEs faster by reusing prior work. The authors tackle two key problems: the tricky sharp behavior of Green's function near certain points, and making sure the network’s predicted values are consistent with their gradients. Their approach learns only the smooth part of the function and ensures consistency, resulting in better accuracy on multiple test problems.
What this means in practice
- •For computational engineers: Efficiently solve PDEs under varying boundary conditions by reusing learned Green's functions for simulations like heat and fluid flow.
- •For software engineers in simulation: Build PDE solvers that integrate learned Green's functions for faster repeated solutions with less reliance on large neural network models.
Authors
Yingchao Huang, Xin Wang, Shanshan Yao, Fanhua Zeng, Wei Peng
Abstract
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 $3.70\times10^{-4}$ with a smaller network architecture, compared with $9.60\times10^{-4}$ for the same-width baseline and $4.63\times10^{-4}$ for a larger configuration, demonstrating that structural improvements are more effective than increasing model size.