A holomorphic Kolmogorov-Arnold network framework for solving elliptic problems on arbitrary 2D domains

TL;DR

This paper introduces a novel holomorphic neural network framework based on Kolmogorov-Arnold representation for solving elliptic PDEs on arbitrary 2D domains, improving accuracy and extending applicability to complex geometries.

Contribution

It presents a new holomorphic network architecture, extends PIHNNs to more elliptic PDEs, and introduces Laurent series-based methods for multiply-connected domains.

Findings

Higher accuracy with reduced model complexity.

Extended applicability to Helmholtz and other elliptic equations.

Ability to handle multiply-connected geometries.

Abstract

Physics-informed holomorphic neural networks (PIHNNs) have recently emerged as efficient surrogate models for solving differential problems. By embedding the underlying problem structure into the network, PIHNNs require training only to satisfy boundary conditions, often resulting in significantly improved accuracy and computational efficiency compared to traditional physics-informed neural networks (PINNs). In this work, we improve and extend the application of PIHNNs to two-dimensional problems. First, we introduce a novel holomorphic network architecture based on the Kolmogorov-Arnold representation (PIHKAN), which achieves higher accuracy with reduced model complexity. Second, we develop mathematical extensions that broaden the applicability of PIHNNs to a wider class of elliptic partial differential equations, including the Helmholtz equation. Finally, we propose a new method based on Laurent series theory that enables the application of holomorphic networks to multiply-connected plane domains, thereby removing the previous limitation to simply-connected geometries.