# Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks

**Authors:** Yizheng Wang, Jia Sun, Jinshuai Bai, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, Yinghua Liu  
**arXiv:** [2406.11045](https://arxiv.org/abs/2406.11045)  
**Date:** 2024-06-16  
**Summary page:** https://www.summarizepaper.com/en/paper/2406.11045v2/

## Key points

- Artificial intelligence for partial differential equations (PDEs) has seen a surge in interest, particularly with the emergence of Physics-informed neural networks (PINNs) and the Kolmogorov-Arnold Network (KAN).
- KANs offer enhanced interpretability and require fewer parameters compared to traditional Multilayer Perceptrons (MLPs), making them attractive for solving PDEs.
- Different formulations of PDEs, such as strong form, energy form, and inverse form, have varying computational implications that are essential to explore in computational physics.
- The Kolmogorov-Arnold-Informed Neural Network (KINN) leverages KAN instead of MLP to tackle forward and inverse problems associated with PDEs.
- Systematic comparisons across various numerical examples show that KINN consistently outperforms MLP in terms of accuracy and convergence speed for diverse PDE scenarios encountered in computational solid mechanics.
- While KINN demonstrates potential for enhancing efficiency and accuracy in solving PDEs within AI frameworks, challenges may arise with complex geometry problems where MLP could perform competitively.

## Summary

In the realm of artificial intelligence for partial differential equations (PDEs), there has been a notable surge in interest. This is especially true with the rise of Physics-informed neural networks (PINNs) and the introduction of the Kolmogorov-Arnold Network (KAN). These advancements have opened up new possibilities for revisiting and enhancing previous MLP-based PINNs. Unlike traditional Multilayer Perceptrons (MLPs), KANs offer enhanced interpretability and require fewer parameters, making them an attractive option for solving PDEs. PDEs can be formulated in various ways, such as strong form, energy form, and inverse form. While these forms may be mathematically equivalent, their computational implications differ significantly. This underscores the importance of exploring different PDE formulations in computational physics. Building upon this understanding, a novel approach is proposed that leverages KAN instead of MLP. This results in the development of the Kolmogorov-Arnold-Informed Neural Network (KINN) for tackling both forward and inverse problems associated with PDEs. To assess the efficacy of KINN compared to MLP, a systematic comparison is conducted across a range of numerical examples encompassing diverse PDE scenarios. These scenarios include multi-scale phenomena, singularities, stress concentrations, nonlinear hyperelasticity behaviors, heterogeneous material properties, and complex geometries. The results obtained from these comparisons showcase that KINN consistently outperforms MLP in terms of accuracy and convergence speed for numerous PDEs encountered in computational solid mechanics. While KINN demonstrates remarkable potential for enhancing efficiency and accuracy in solving PDEs within AI frameworks, it is worth noting that there are certain challenges posed by complex geometry problems where MLP may exhibit competitive performance. Nevertheless, the overall findings underscore the promise of KINN as a powerful tool for advancing AI solutions for PDEs across various domains within computational physics.