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

2026-01-16

AI Summary

This work proposes a novel framework for efficiently learning solution operators of linear partial differential equations using only boundary data—specifically, solution values and normal derivatives. By leveraging fundamental solutions to construct physically consistent synthetic data (termed Mathematical Artificial Data, or MAD) and combining it with boundary integral representations, the method enables accurate reconstruction of interior solutions without requiring full-domain sampling or numerical simulations. For the first time, MAD is integrated with boundary neural operators to establish a purely data-driven boundary-to-boundary mapping. The approach uniformly handles Dirichlet, Neumann, and mixed boundary conditions, achieving accuracy comparable to or better than existing neural operators on two-dimensional Laplace, Poisson, and Helmholtz equations, while significantly improving training efficiency. Moreover, it naturally extends to three-dimensional domains and complex geometries.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• operator learning
• boundary data
• linear partial differential equations
• fundamental solutions
• boundary integral

Innovation

• operator learning
• boundary integral
• fundamental solution
• artificial data
• neural operator