Neural Green's Functions

2025-11-02

AI Summary

Generalizing neural operators for linear partial differential equations (PDEs) on irregular geometries to unseen source terms and boundary conditions remains challenging due to mesh dependency and lack of physical priors. Method: We propose Neural Green’s Function (NGF), the first neural operator that embeds the physical prior of Green’s functions via spectral decomposition, enabling decoupled generalization over sources and boundary conditions without meshing. NGF processes voxelized point clouds to extract local geometric features, predicts spectral coefficients of the Green’s operator, and computes solutions efficiently via numerical integration. Contribution/Results: On the MCB mechanical part thermal analysis benchmark, NGF reduces mean error by 13.9% over the best baseline neural operator and accelerates inference up to 350× compared to traditional numerical solvers. It demonstrates superior generalization across diverse geometries and functional forms of sources and boundary conditions.

Technology Category

• Machine Learning: Efficient ML / Green AI
• Search and Optimization: Mixed Discrete/Continuous Search
• Natural Language Processing: Language Grounding & Multi-modal NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Solving linear PDEs with neural operators for irregular geometries
• Improving generalization across unseen source and boundary functions
• Accelerating thermal analysis of mechanical parts compared to numerical solvers

Innovation

• Neural operator imitates Green's functions for PDEs
• Extracts point features to predict solution decomposition
• Achieves geometry-agnostic generalization via numerical integration