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

arXiv: https://arxiv.org/abs/2605.31231 · HTML: https://mlsanity.com/paper/2605.31231/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 80 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Minimum Width for Universal Approximation](https://mlsanity.com/paper/2006.08859/) (2020)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios](https://mlsanity.com/paper/2205.07731/) (2022)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Achieve the Minimum Width of Neural Networks for Universal Approximation](https://mlsanity.com/paper/2209.11395/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximation](https://mlsanity.com/paper/2305.18460/) (2023)
- [A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity](https://mlsanity.com/paper/2108.07243/) (2021)
- [BINN: A deep learning approach for computational mechanics problems based on boundary integral equations](https://mlsanity.com/paper/2301.04480/) (2023)
- [Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving Linear Elasticity Problems](https://mlsanity.com/paper/2407.01088/) (2024)
- [DCEM: A deep complementary energy method for solid mechanics](https://mlsanity.com/paper/2302.01538/) (2023)
- [A Stabilised Nodal Spectral Element Method for Fully Nonlinear Water Waves](https://mlsanity.com/paper/1512.02548/) (2015)
- [A Neural Solver for Variational Problems on CAD Geometries with Application to Electric Machine Simulation](https://mlsanity.com/paper/2111.09005/) (2021)
- [Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activation](https://mlsanity.com/paper/2203.13410/) (2022)
- [A holomorphic Kolmogorov-Arnold network framework for solving elliptic problems on arbitrary 2D domains](https://mlsanity.com/paper/2507.22678/) (2025)
- [Transfer-learned Kolosov-Muskhelishvili Informed Neural Networks for Fracture Mechanics](https://mlsanity.com/paper/2601.00491/) (2026)
- [Elimination-compensation pruning for fully-connected neural networks](https://mlsanity.com/paper/2602.20467/) (2026)

## Built on by (most cited first, 2 of 2)

- [AURA: Angular Update Rate Adaptation for training complex-valued neural networks](https://mlsanity.com/paper/2609.25914/) (2026)
- [Enriched Holomorphic Neural Networks for 2D and 3D Laplace Problems](https://mlsanity.com/paper/2610.12541/) (2026)

## Abstract (arXiv)

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.