Neural Green's Functions

TL;DR

Neural Green's Function is a neural operator designed for linear PDEs that generalizes well across irregular geometries and source functions, outperforming existing methods in accuracy and speed.

Contribution

We propose Neural Green's Function, a novel neural solution operator inspired by Green's functions, capable of robustly generalizing across diverse geometries and functions.

Findings

Achieves 13.9% error reduction over state-of-the-art neural operators.

Outperforms traditional solvers by up to 350 times in speed.

Demonstrates superior generalization to unseen geometries and functions.

Abstract

We introduce Neural Green's Function, a neural solution operator for linear partial differential equations (PDEs) whose differential operators admit eigendecompositions. Inspired by Green's functions, the solution operators of linear PDEs that depend exclusively on the domain geometry, we design Neural Green's Function to imitate their behavior, achieving superior generalization across diverse irregular geometries and source and boundary functions. Specifically, Neural Green's Function extracts per-point features from a volumetric point cloud representing the problem domain and uses them to predict a decomposition of the solution operator, which is subsequently applied to evaluate solutions via numerical integration. Unlike recent learning-based solution operators, which often struggle to generalize to unseen source or boundary functions, our framework is, by design, agnostic to the specific functions used during training, enabling robust and efficient generalization. In the steady-state thermal analysis of mechanical part geometries from the MCB dataset, Neural Green's Function outperforms 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 a numerical solver that requires computationally expensive meshing.