Multi-Resolution Training-Enhanced Kolmogorov-Arnold Networks for Multi-Scale PDE Problems
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913960750481408 |
|---|---|
| author | Yang, Yu-Sen Guo, Ling Ren, Xiaodan |
| author_facet | Yang, Yu-Sen Guo, Ling Ren, Xiaodan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19888 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-Resolution Training-Enhanced Kolmogorov-Arnold Networks for Multi-Scale PDE Problems Yang, Yu-Sen Guo, Ling Ren, Xiaodan Computational Physics 65M32 G.1.8 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. |
| title | Multi-Resolution Training-Enhanced Kolmogorov-Arnold Networks for Multi-Scale PDE Problems |
| topic | Computational Physics 65M32 G.1.8 |
| url | https://arxiv.org/abs/2507.19888 |