A Hybrid Kernel-Free Boundary Integral Method with Operator Learning for Solving Parametric Partial Differential Equations In Complex Domains

The Kernel-Free Boundary Integral (KFBI) method presents an iterative solution to boundary integral equations arising from elliptic partial differential equations (PDEs). This method effectively addresses elliptic PDEs on irregular domains, including the modified Helmholtz, Stokes, and elasticity equations. The rapid evolution of neural networks and deep learning has invigorated the exploration of numerical PDEs. An increasing interest is observed in deep learning approaches that seamlessly integrate mathematical principles for investigating numerical PDEs. We propose a hybrid KFBI method, integrating the foundational principles of the KFBI method with the capabilities of deep learning. This approach, within the framework of the boundary integral method, designs a network to approximate the solution operator for the corresponding integral equations by mapping the parameters, inhomogeneous terms and boundary information of PDEs to the boundary density functions, which can be regarded as the solution of the integral equations. The models are trained using data generated by the Cartesian grid-based KFBI algorithm, exhibiting robust generalization capabilities. It accurately predicts density functions across diverse boundary conditions and parameters within the same class of equations. Experimental results demonstrate that the trained model can directly infer the boundary density function with satisfactory precision, obviating the need for iterative steps in solving boundary integral equations. Furthermore, applying the inference results of the model as initial values for iterations is also reasonable; this approach can retain the inherent second-order accuracy of the KFBI method while accelerating the traditional KFBI approach by reducing about 50% iterations.

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

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.