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

arXiv: https://arxiv.org/abs/2601.11222 · HTML: https://mlsanity.com/paper/2601.11222/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 34 references that are in the map.
- First posted on arXiv 16 Jan 2026.

## Builds on (most cited first, 9 of 9)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solving](https://mlsanity.com/paper/2408.02198/) (2024)
- [BI-GreenNet: Learning Green's functions by boundary integral network](https://mlsanity.com/paper/2204.13247/) (2022)
- [Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems](https://mlsanity.com/paper/2412.03161/) (2024)
- [Mathematical artificial data for operator learning](https://mlsanity.com/paper/2507.06752/) (2025)

## Abstract (arXiv)

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.