An explainable operator approximation framework under the guideline of Green's function

2024-12-21

arXiv.org

AI Summary

To address the high computational cost of traditional PDE solvers (e.g., FEM/FVM) and the difficulty of analytically deriving Green’s functions, this paper proposes an interpretable deep operator learning framework grounded in the Green’s function integral representation. Our method decouples modeling into two neural subnetworks: a Trunk Net that learns the implicit Green’s kernel, and a Branch Net that encodes source/boundary inputs and their gradient responses. Crucially, we introduce an embedded Green’s function learning mechanism, where the integral operator is parameterized via physics-informed neural networks to tightly couple boundary and source terms. This approach directly learns the Green’s function and its spatial gradients without mesh discretization, enhancing both efficiency and generalizability. Extensive experiments on 3D heat conduction, reaction–diffusion, and Stokes equations demonstrate superior accuracy, extrapolation capability, and interpretability compared to PINNs, DeepONet, PI-DeepONet, and FNO.

Technology Category

• Machine Learning: Efficient ML / Green AI
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

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

Abstract

Traditional numerical methods, such as the finite element method and finite volume method, adress partial differential equations (PDEs) by discretizing them into algebraic equations and solving these iteratively. However, this process is often computationally expensive and time-consuming. An alternative approach involves transforming PDEs into integral equations and solving them using Green's functions, which provide analytical solutions. Nevertheless, deriving Green's functions analytically is a challenging and non-trivial task, particularly for complex systems. In this study, we introduce a novel framework, termed GreensONet, which is constructed based on the strucutre of deep operator networks (DeepONet) to learn embedded Green's functions and solve PDEs via Green's integral formulation. Specifically, the Trunk Net within GreensONet is designed to approximate the unknown Green's functions of the system, while the Branch Net are utilized to approximate the auxiliary gradients of the Green's function. These outputs are subsequently employed to perform surface integrals and volume integrals, incorporating user-defined boundary conditions and source terms, respectively. The effectiveness of the proposed framework is demonstrated on three types of PDEs in bounded domains: 3D heat conduction equations, reaction-diffusion equations, and Stokes equations. Comparative results in these cases demonstrate that GreenONet's accuracy and generalization ability surpass those of existing methods, including Physics-Informed Neural Networks (PINN), DeepONet, Physics-Informed DeepONet (PI-DeepONet), and Fourier Neural Operators (FNO).

Problem

• Approximates Green's functions for PDEs efficiently
• Solves PDEs using integral formulations with neural networks
• Outperforms existing methods in accuracy and generalization

Innovation

• GreensONet learns embedded Green's functions
• Trunk Net approximates unknown Green's functions
• Branch Net approximates auxiliary gradients