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Learning Domain-Independent Green's Function For Elliptic Partial Differential Equations

Pawan Negi, Maggie Cheng, Mahesh Krishnamurthy, Wenjun Ying, Shuwang Li

TL;DR

This work introduces BIN-G, a boundary-integral neural network that learns a domain-independent Green's function $G(\mathbf{x}, \mathbf{y})$ for elliptic PDEs by representing $G$ with a radial-basis-Kernel network and jointly learning boundary densities via MLPs. The method enforces Green's function symmetry and handles singularities by training on a 1D radial domain for the PDE residual, while boundary-integral losses on prescribed test functions guide the densities, enabling accurate solutions across diverse domains and boundary conditions. Demonstrations on 2D Laplace and Helmholtz equations, as well as a variable-coefficient PDE, show that the learned GF generalizes to new domains and coefficients, with domain-shaped errors typically in the low single digits and acceptable performance even near corners. The approach offers a domain-agnostic, computationally efficient pathway to apply boundary-integral formulations in heterogeneous media and moving-interface contexts, reducing reliance on analytic Green's functions and enabling rapid adaptation to new geometries.

Abstract

Green's function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green's function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green's function, referred to as BIN-G. We evaluate the Green's function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green's function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green's function for PDEs with variable coefficients. The learned Green's function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.

Learning Domain-Independent Green's Function For Elliptic Partial Differential Equations

TL;DR

This work introduces BIN-G, a boundary-integral neural network that learns a domain-independent Green's function for elliptic PDEs by representing with a radial-basis-Kernel network and jointly learning boundary densities via MLPs. The method enforces Green's function symmetry and handles singularities by training on a 1D radial domain for the PDE residual, while boundary-integral losses on prescribed test functions guide the densities, enabling accurate solutions across diverse domains and boundary conditions. Demonstrations on 2D Laplace and Helmholtz equations, as well as a variable-coefficient PDE, show that the learned GF generalizes to new domains and coefficients, with domain-shaped errors typically in the low single digits and acceptable performance even near corners. The approach offers a domain-agnostic, computationally efficient pathway to apply boundary-integral formulations in heterogeneous media and moving-interface contexts, reducing reliance on analytic Green's functions and enabling rapid adaptation to new geometries.

Abstract

Green's function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green's function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green's function, referred to as BIN-G. We evaluate the Green's function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green's function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green's function for PDEs with variable coefficients. The learned Green's function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.
Paper Structure (13 sections, 1 theorem, 34 equations, 13 figures, 1 algorithm)

This paper contains 13 sections, 1 theorem, 34 equations, 13 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Boundary integral network architecture
  4. Green's function network architecture
  5. BIN-G training methodology
  6. Green's function training scheme
  7. Generalization to other domain and boundary conditions
  8. Results and discussion
  9. Laplace equation
  10. Helmholtz equation
  11. PDE with variable coefficient
  12. Conclusions
  13. Acknowledgements

Key Result

Theorem 1

Given a PDE of the form $\mathscr{L}_\mathbf{x} u(\mathbf{x}) = f(\mathbf{x}), \forall \ \mathbf{x} \in \Omega$ with a non-homogenous boundary condition, the solution of the interior Dirichlet problem is given by with where $\mathbf{x}_o^-$ mean converging in the interior of $\Omega$. Similarly, for the interior Neumann problem, the solution is given by with

Figures (13)

  • Figure 1: Network architecture of the BIN-G. Using the distance $r_i$ between $\mathbf{x}$ and $\mathbf{x}_i^b$ enforcing the symmetry of the Green's function. The block highlighted in orange evaluates a domain-independent Green's function learned using a KNN.
  • Figure 2: KNN-based architecture to evaluate domain-independent Green's function.
  • Figure 3: Comparison of the approximation of the Dirac delta and the expected derivative of the Green's function at the singularity.
  • Figure 4: The sample space used in the training of the domain independent Green's function network. The domain used to compute the loss due to the PDE is a 1D domain shown in black, representing the domain in blue for a 2D PDE (due to symmetry of the Green's function). The orange domain is used to compute loss due to boundary integral equations.
  • Figure 5: Comparison of the learned and analytical Green's functions and convergence of $L_2$ error in the test data while training density function using learned and analytical Green's function of Laplace equation.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Definition 1
  • Theorem 1