Operator learning on domain boundary through combining fundamental solution-based artificial data and boundary integral techniques

TL;DR

This work addresses operator learning for linear PDEs with known fundamental solutions by learning a boundary-to-boundary map using synthetic, physics-embedded boundary data. It combines the integral representation with a fundamental-solution–based data generation strategy (MAD) to train a boundary-aware neural operator (MAD-BNO) that maps Dirichlet and Neumann data on complementary boundary segments to their unknown counterparts, enabling the interior solution to be recovered via boundary integrals. The approach achieves comparable or superior accuracy to existing neural-operator methods while dramatically reducing training time, and it supports Dirichlet, Neumann, and mixed boundaries as well as general sources for Laplace, Poisson, and Helmholtz equations; it also extends naturally to three dimensions. By shifting learning to the boundary and leveraging physics-informed data, MAD-BNO offers a mesh-free, geometry-flexible framework with potential real-time applicability and scalable performance in higher dimensions.

Abstract

For linear partial differential equations with known fundamental solutions, this work introduces a novel operator learning framework that relies exclusively on domain boundary data, including solution values and normal derivatives, rather than full-domain sampling. By integrating the previously developed Mathematical Artificial Data (MAD) method, which enforces physical consistency, all training data are synthesized directly from the fundamental solutions of the target problems, resulting in a fully data-driven pipeline without the need for external measurements or numerical simulations. We refer to this approach as the Mathematical Artificial Data Boundary Neural Operator (MAD-BNO), which learns boundary-to-boundary mappings using MAD-generated Dirichlet-Neumann data pairs. Once trained, the interior solution at arbitrary locations can be efficiently recovered through boundary integral formulations, supporting Dirichlet, Neumann, and mixed boundary conditions as well as general source terms. The proposed method is validated on benchmark operator learning tasks for two-dimensional Laplace, Poisson, and Helmholtz equations, where it achieves accuracy comparable to or better than existing neural operator approaches while significantly reducing training time. The framework is naturally extensible to three-dimensional problems and complex geometries.

Figures (20)

• Figure 1: Mapping a general triangle $T$ with vertices $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ to the reference triangle $\hat{T}$.
• Figure 2: Training loss curve for the Laplace equation (Eq. \ref{['eq:laplace']}) with Dirichlet boundary conditions. The vertical axis is shown on a logarithmic scale to emphasize the convergence rate of the model. The plotted loss values represent the average loss within every 100 epochs to better illustrate the training progress.
• Figure 3: Comparison of MAD-BNO and PI-DeepONet's predictions for the analytical solution $\bm{u(x, y) = \ln\left( (x - 3)^2 + (y - 4)^2 \right)}$ of the Laplace equation in the square domain $(0,1) \times (0,1)$. (g): Comparison of the Neumann boundary values predicted by MAD-BNO with the exact values, where the square boundary of the domain is parameterized into the interval $[0,4]$ by traversing counterclockwise from the point $(0,0)$.
• Figure 4: Comparison of MAD-BNO and PI-DeepONet's predictions for the analytical solution $\bm {u(x, y) = \ln\left( (x + 3)^2 + (y - 4)^2 \right)}$ of the Laplace equation in a domain enclosed by the polar curve $r(\theta) =1$, with $\theta \in [0, 2\pi)$.
• Figure 5: Comparison of MAD-BNO and PI-DeepONet's predictions for the analytical solution $\bm{ u(x, y) =sin(x+1)sinh(y) }$ of the Laplace equation in a domain enclosed by the polar curve $r_{1}(\theta) = 0.8+0.1sin(2\theta)$ and $r_{2}(\theta) = 0.3+0.05sin(2\theta)+0.03sin(3\theta)$, with $\theta \in [0, 2\pi)$.