[Paper Review] BI-GreenNet: Learning Green's functions by boundary integral network

Guochang Lin, Fukai Chen+5 more|arXiv (Cornell University)|Apr 28, 2022

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes BI-GreenNet, a neural network method that leverages boundary integral formulations and fundamental solutions to learn Green’s functions for Poisson and Helmholtz equations in bounded, unbounded, and interface domains. By reformulating the problem into a smooth boundary-only fitting task, BI-GreenNet achieves high-accuracy Green’s function approximation and enables generalization to solve diverse PDEs with relative $ L^2 $ errors below 4%.

ABSTRACT

Green's function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green's function is difficult to compute. The troubles arise in the following three folds. Firstly, compared with the original PDE, the dimension of Green's function is doubled, making it impossible to be handled by traditional mesh-based methods. Secondly, Green's function usually contains singularities which increase the difficulty to get a good approximation. Lastly, the computational domain may be very complex or even unbounded. To override these problems, we leverage the fundamental solution, boundary integral method and neural networks to develop a new method for computing Green's function with high accuracy in this paper. We focus on Green's function of Poisson and Helmholtz equations in bounded domains, unbounded domains. We also consider Poisson equation and Helmholtz domains with interfaces. Extensive numerical experiments illustrate the efficiency and the accuracy of our method for solving Green's function. In addition, we also use the Green's function calculated by our method to solve a class of PDE, and also obtain high-precision solutions, which shows the good generalization ability of our method on solving PDEs.

Motivation & Objective

• Address the challenge of computing Green’s functions for PDEs in complex or unbounded domains where traditional methods fail due to high dimensionality and singularities.
• Overcome the limitations of mesh-based methods and standard neural networks in handling the doubled dimension and non-smooth nature of Green’s functions.
• Develop a differentiable framework that leverages fundamental solutions and boundary integral representations to enable accurate and generalizable Green’s function learning.
• Demonstrate the method’s ability to generalize to unseen PDEs by using the learned Green’s function to solve new source-term and boundary condition problems with high precision.

Proposed method

• Reformulate the Green’s function problem into a boundary integral equation using the fundamental solution, which removes the singularity and transforms the PDE into a smooth, high-dimensional function approximation task.
• Propose BI-GreenNet, a neural network architecture that parameterizes the Green’s function in a boundary integral form, ensuring the PDE is automatically satisfied and only boundary conditions are trained.
• Use a neural network to approximate the unknown boundary density function in the boundary integral representation, with loss functions based on boundary conditions and residual minimization.
• Integrate automatic differentiation and physics-informed training to optimize the network weights using both PDE residuals and boundary constraints.
• Compare BI-GreenNet with DB-GreenNet, a direct approximation method, to validate the superiority of the boundary integral formulation in accuracy and scalability.
• Leverage the learned Green’s function to solve new PDEs via convolution-like integration, demonstrating generalization across source terms and boundary conditions.

(a) Single domain

Experimental results

Research questions

• RQ1 Can a neural network effectively learn Green’s functions for Poisson and Helmholtz equations in complex domains, including unbounded and interface domains?
• RQ2 Does the boundary integral formulation in BI-GreenNet outperform direct neural network approximation (DB-GreenNet) in accuracy and stability, especially for singular or high-dimensional problems?
• RQ3 To what extent can a learned Green’s function generalize to solve new PDEs with different source terms and boundary conditions?
• RQ4 How does the use of fundamental solutions and boundary integral representations mitigate the challenges of dimensionality doubling and singularity in Green’s function computation?
• RQ5 Can the method maintain high accuracy in unbounded domains where traditional mesh-based methods are infeasible?

Key findings

• BI-GreenNet achieves high-accuracy learning of Green’s functions for Poisson and Helmholtz equations in bounded, unbounded, and interface domains, outperforming the direct DB-GreenNet approach.
• The method successfully handles singularities and high-dimensional problems by reformulating the Green’s function as a boundary integral, enabling stable and accurate training.
• In unbounded domains, BI-GreenNet maintains superior performance compared to DB-GreenNet, demonstrating robustness to domain complexity.
• The learned Green’s function generalizes well to solve new PDEs: when applied to 100 randomly generated interface problems, all relative $ L^2 $ errors remained below 4%.
• The method enables accurate solution of PDEs via boundary integral evaluation using the learned Green’s function, confirming its practical utility and generalization capacity.
• The framework is effective even for problems with complex geometries and discontinuous coefficients, as validated through interface problems with varying material parameters.

(b) Interface problem