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

TL;DR

This paper introduces MAD-BNO, a boundary-focused operator learning method that synthesizes training data from fundamental solutions, enabling efficient and accurate solutions for PDEs using boundary data alone.

Contribution

It presents a novel boundary-only operator learning framework combining MAD and boundary integral techniques, reducing data requirements and training time for PDE solutions.

Findings

Achieves accuracy comparable or better than existing neural operators.

Reduces training time significantly.

Supports complex geometries and 3D problems.

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.