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

Abstract

AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN) for solving forward and inverse problems. We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP regarding accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

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• A Practitioner's Guide to Kolmogorov-Arnold Networks
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• Training Deep Physics-Informed Kolmogorov-Arnold Networks
• Variational PINNs with tree-based integration and boundary element data in the modeling of multi-phase architected materials
• Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging
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• CVKAN: Complex-Valued Kolmogorov-Arnold Networks
• Unraveling particle dark matter with Physics-Informed Neural Networks
• QKAN: quantum Kolmogorov-Arnold networks with applications in machine learning and multivariate state preparation
• CompNO: A Novel Foundation Model approach for solving Partial Differential Equations
• Lightweight posterior construction for gravitational-wave catalogs with the Kolmogorov-Arnold network
• KAN-Enhanced Contrastive Learning Accelerating Crystal Structure Identification from XRD Patterns