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

2025-07-30

AI Summary

This work addresses the efficient and high-accuracy solution of elliptic partial differential equations (e.g., Laplace and Helmholtz equations) on arbitrary two-dimensional domains—including both simply and multiply connected ones. We propose the Physics-Informed Holomorphic Kolmogorov–Arnold Network (PIHKAN), which integrates complex analysis theory with the Kolmogorov–Arnold representation theorem. By explicitly incorporating Laurent series expansions, PIHKAN models topological singularities inherent in multiply connected domains—overcoming a key limitation of conventional holomorphic neural networks restricted to simply connected domains. Furthermore, PIHKAN embeds governing PDE physics and boundary conditions directly into the architecture, enabling mesh-free, boundary-driven, end-to-end training. Experiments demonstrate that PIHKAN achieves substantial improvements in solution accuracy and generalization robustness while significantly reducing model complexity. This work establishes a novel paradigm for intelligent, physics-aware solving of elliptic PDEs in complex domains.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Application Domains: Humanities & Computational Social Science

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Develop holomorphic network for 2D elliptic problems
• Extend PIHNNs to broader elliptic PDE classes
• Enable holomorphic networks on multiply-connected domains

Innovation

• Holomorphic network based on Kolmogorov-Arnold representation
• Mathematical extensions for broader elliptic PDEs
• Laurent series method for multiply-connected domains