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

TL;DR

This work addresses solving parametric PDEs in complex domains by coupling the Kernel-Free Boundary Integral (KFBI) method with neural operator learning to map boundary data and PDE parameters to the boundary density $\varphi$. The approach learns a solution operator $\mathcal{S}_{\mathcal{L},\Omega}$ to predict $\varphi$ directly (Strategy 1) or to provide a high-quality initial guess for KFBI iterations (Strategy 2), achieving substantial reductions in iteration counts while maintaining second-order accuracy. It demonstrates robust generalization across Laplace, Poisson, Stokes, and Modified Helmholtz problems on both fixed and parametric domains, with notable speedups (often exceeding 50%) and accuracy around $10^{-3}$ to $10^{-4}$ in many cases. The results suggest strong potential for integrating the hybrid KFBI framework with HPC and GPU architectures, and point to fruitful future directions including time-dependent PDEs and 3D extensions for engineering applications.

Abstract

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.

Figures (6)

• Figure 1: BI method and KFBI method computation domain
• Figure 2: Network architecture schematic for networks with parameter component. In this graph, $(I_{1, 1}, I_{1, 2}, ..., I_{1, M})^{T} = I_1$ and $(I_{2, 1}, I_{2, 2}, ..., I_{2, M})^{T} = I_2$ are used to present the intermediate elements, $\widetilde{g}_i$ is used to present $\widetilde{g_D}(\mathbf{x_i})$ and $\varphi_i$ is used to present $\varphi(\mathbf{x_i})$ for all $i = 1, 2, ..., M$. The schematic diagram of fully connected networks do not represent the actual size of the network model in practical use, they are for illustrative purposes only and these three blocks do not share their parameters. Also note that the 'Flatten' and 'reshape' operations are performed to obtain the correct shapes of tensors, enabling them to be properly passed to the next network block, and some terms in this diagram have the same meanings as their corresponding terms in the PyTorch library, such as 'Conv2d' represents 2D convolution and 'ConvTransposed2d' represents 2D transposed convolution.
• Figure 3: Numerical solutions of Modified Helmholtz equation 1 given by Strategy 1 and Strategy 2 in grid $256 \times 256$. Note that the figures show the results given by solving the corresponding interface problem in KFBI whose interior solution is the desired results for the original PDE.
• Figure 4: Exact solution and Numerical solution of Modified Helmholtz equation 2 given by Strategy 1 in grid $128 \times 128$. Note that the right figure show the result given by solving the corresponding interface problem in KFBI whose interior solution is the desired result for the original PDE.
• Figure 5: Numerical solution of the 2D Naiver equation 1 given by Strategy 2 in grid $256 \times 256$. Note that the figures show the result given by solving the corresponding interface problem in KFBI whose interior solution is the desired result for the original PDE.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7