Multi-resolution training-enhanced Kolmogorov-Arnold networks for physics-informed learning of PDEs with multi-scale features

Abstract

PDE problems with multi-scale features present significant challenges in scientific computing. While conventional deep learning methods based on multi-layer perceptrons exhibit spectral bias in resolving multi-scale features, physics-informed Kolmogorov-Arnold networks mitigate this issue through their architectural design and show advantages in representing multi-scale physical fields. On the other hand, insights from information bottleneck theory suggest that high-resolution sampling is important for capturing multi-scale behavior, but this requirement often leads to increased computational cost. To address this issue, we propose a simple yet effective multi-resolution training-enhanced Kolmogorov-Arnold network framework, which trains the model sequentially or alternately at different resolutions. The proposed framework is validated on various forward and inverse PDE problems with multi-scale features. Numerical results indicate that this training strategy effectively reduces computational cost without sacrificing accuracy, thereby enabling the efficient solution of complex multi-scale PDE problems in both forward and inverse settings.