Multi-Resolution Training-Enhanced Kolmogorov-Arnold Networks for Multi-Scale PDE Problems

2025/07/26 by Yusen Yang, Ling Guo, Yang, Yu-Sen +3 · 1 citation
Computer Science · Materials Science · Physics and Astronomy · #65M32 #Computational Physics (physics.comp-ph) #FOS: Physical sciences #G.1.8 #Machine Learning in Materials Science #Model Reduction and Neural Networks #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.2507.19888

openalex publication_date 2025/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Multi-scale PDE problems present significant challenges in scientific computing. While conventional MLP-based deep learning methods exhibit spectral bias in resolving multi-scale features, the physics-informed Kolmogorov-Arnold network (PIKAN) mitigates this issue through its novel architecture, demonstrating certain advantages. On the other hand, insights from the information bottleneck theory suggest that high-resolution training points are essential for these hybrid methods to accurately capture multi-scale behavior, although this requirement often leads to longer training times. To address this challenge, we propose a simple yet effective multi-resolution training-enhanced PIKAN framework, termed MR-PIKAN, which trains the data-physics hybrid model either sequentially or alternately across different resolutions. The proposed MR-PIKAN is validated on various multi-scale forward and inverse PDE problems. Numerical results indicate that this new training strategy effectively reduces computational costs without sacrificing accuracy, thereby enabling efficient solutions of complex multi-scale PDEs in both forward and inverse settings.

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Cited by

• Towards a Foundation Model for Partial Differential Equations Across Physics Domains

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