A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials

2026-05-29

AI Summary

This work addresses the challenge of simultaneously achieving high accuracy, mesh-free flexibility, and structural fidelity to the governing equations in solving three-dimensional harmonic potential boundary-value problems. The authors propose a novel approach that leverages the Whittaker integral representation to express the solution as a holomorphic function of complex variables, which is then approximated using a holomorphic neural network. By training solely through boundary collocation—without requiring domain-based PDE residual terms—the method constructively satisfies the governing Laplace equation exactly. This is the first integration of holomorphic neural networks with harmonic potential theory, enabling highly accurate, globally controllable solutions for both scalar and vector fields in three-dimensional Laplace and linear elasticity problems, thereby significantly enhancing the accuracy and robustness of mesh-free deep learning methods.

Abstract

We present a neural-network-based framework for the solution of three-dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, which allows representing the solution through functions that are holomorphic with respect to a suitable complex variable. These functions are subsequently approximated using holomorphic neural networks, which guaranty fulfillment of the holomorphicity requirement. A key feature of the proposed formulation is that the governing partial differential equations (PDEs) are satisfied exactly by construction. Therefore, in contrast to standard physics-informed neural networks, no residual minimization of PDEs is required in the interior of the domain, and training is based exclusively on boundary collocation points. The method is validated against three-dimensional Laplace and linear elasticity problems, where, in the latter case, displacement and stress fields are expressed via the Papkovich-Neuber potentials. The numerical results show an accurate approximation of both scalar and vector fields, with errors remaining controlled throughout the domain. Overall, the work demonstrates that the incorporation of analytical structures into neural network architectures provides a natural and effective framework for the meshless approximation of three-dimensional boundary value problems while preserving the underlying properties of the governing equations.

Problem

• boundary value problems
• harmonic potentials
• holomorphic functions
• 3D Laplace equation
• linear elasticity

Innovation

• holomorphic neural networks
• boundary value problems
• harmonic potentials
• Whittaker integral formula
• meshless approximation