Neural Green’s Functions for Solving Poisson’s Equation
Seungwoo Yoo, Kyeongmin Yeo, Jisung Hwang, Minhyuk Sung
We introduce Neural Green’s Function, a neural solution operator for Poisson’s equation that achieves superior generalization to diverse irregular geometries and functions. The foundation of our approach is inspired by Green’s functions, which serve as solution operators for linear PDEs, including Poisson’s equation, and depend exclusively on the geometry of problem domains. Building on the eigendecompositions of Green’s functions, Neural Green’s Function extracts per-point features from a volumetric point cloud representing the problem domain and uses them to predict the terms required for the numerical integration to obtain the solution, enabling robust and efficient handling of diverse domains, source functions, and boundary conditions. Unlike recent learning-based solution operators, which often struggle with unseen source functions and boundary functions, Neural Green’s Function is agnostic to specific source and boundary functions used for training by design. By embedding the property of Green’s functions into the framework design, Neural Green’s Function enables data-efficient learning of solution operators. We demonstrate the superior performance of our method in steady-state thermal analysis of mechanical part shapes from the MCB dataset given unseen source and boundary functions. Neural Green’s Function outperforms the current state-of-the-art neural operators, achieving an average error reduction of 13.9% across five shape categories, while being up to 350 times faster than numerical solvers that rely on computationally expensive mesh generation.