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

TL;DR

This paper proposes KINN, a physics-informed neural framework that substitutes MLPs with Kolmogorov–Arnold Networks (KAN) to solve forward and inverse PDEs expressed in strong, energy, and inverse forms. By learning activation functions via B-splines within KAN and aligning with FEM/IGA concepts, KINN achieves higher accuracy and faster convergence than conventional PINNs in many solid-mechanics PDEs, including problems with singularities, stress concentration, nonlinear hyperelasticity, and heterogeneity. The work provides NTK-based insight into KAN’s reduced spectral bias and highlights KINN’s advantages for multi-scale and highly heterogeneous problems, while also acknowledging limitations on complex geometries and current computational efficiency. Overall, KAN-based approaches are positioned as a promising direction for efficient, accurate AI-driven PDE solvers, with future improvements focused on geometry handling, mesh adaptation, and weak-form integration.

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.

Figures (27)

• Figure 1: Schematic of KINN: The idea of KINN is to replace MLP with Kolmogorov–Arnold Networks (KAN) in different PDEs forms (strong, energy, and inverse forms). In KAN, the main training parameters are the undetermined coefficients $c_{i}$ of the B-splines in the activation function. KINN establishes the loss function based on different numerical formats of PDEs and optimizes the $c_{i}$. The virtual Grid is determined by the grid size in KAN.
• Figure 2: Validation of the "spectral bias" of MLP and KAN in fitting $u = \sin(2\pi x) + 0.1\sin(50\pi x)$. (a) Exact solutions, MLP predictions, and KAN predictions at epochs 100, 1000, and 10000, (b) Eigenvalue distributions of MLP and KAN, (c) The First row is the eigenvectors of the largest eigenvalue sorted from largest to smallest, and the second row is the eigenvectors of the three smallest eigenvalues, (d) Evolution of the loss functions of MLP and KAN.
• Figure 3: MLP and KAN fitting the high and low-frequency mixed heat conduction problem. (a) Exact solution for frequency $F=50$. (b) MLP prediction for frequency $F=50$ after 3000 iterations. (c) The absolute error of MLP for frequency $F=50$ after 3000 iterations. (d) KAN prediction for frequency $F=50$ after 3000 iterations. (e) The absolute error of KAN for frequency $F=50$ after 3000 iterations. (f) Relative error of KAN under different grid sizes and frequencies, with 3000 iterations and network structure [2,5,1].
• Figure 4: Introduction to mode III crack: (a) The structure of the mode III crack, with a size of $\text{[-1,1]}^{2}$ in a square region. $\theta$ is from $[-\pi, \pi]$. The blue and yellow regions represent two neural networks since the displacement is discontinuous at the crack ($x<0,y=0$). Therefore, two neural networks are needed to fit the displacement above and below the crack. (b) The analytical solution for this problem: $u=r^{\frac{1}{2}} \sin(\theta/2)$, where $r$ is the distance from the coordinate $\boldsymbol{x}$ to the origin $\text{x=y=0}$, and $\theta$ is the angle in the polar coordinate system, with $x>0,y=0$ as the reference. Counterclockwise is considered a positive angle. (c) The grid distribution in KINN, with order=3 and grid size=10, is uniformly distributed in both the x and y directions. (d) The meshless random sampling points for PINN, DEM, and KINN. The red points are essential boundary displacement points (256 points), the blue points are for the upper region neural network (2048 points), the green points are for the lower region neural network (2048 points), and the yellow points are the interface sampling points (1000 points).
• Figure 5: The absolute error by CPINN, DEM, BINN, and KINN in mode III crack: CPINN_MLP, DEM_MLP, BINN_MLP respectively (first row), KINN_CPINN, KINN_DEM, KINN_BINN respectively (second row).