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

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.