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

Yizheng Wang, Jia Sun+6 more|arXiv (Cornell University)|Jun 16, 2024

Neural Networks and Applications4 citations

TL;DR

This paper introduces Kolmogorov-Arnold-Informed Neural Networks (KINN), a physics-informed deep learning framework that replaces traditional multilayer perceptrons (MLPs) with Kolmogorov-Arnold Networks (KANs) to solve forward and inverse problems in partial differential equations (PDEs). KINN leverages KANs' learnable activation functions and univariate function decomposition to achieve superior accuracy and convergence speed across diverse PDE problems in computational solid mechanics, except in complex geometries where performance is comparable to MLPs.

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.

Motivation & Objective

• To address the limitations of traditional Physics-Informed Neural Networks (PINNs) based on multilayer perceptrons (MLPs), which suffer from poor interpretability and high parameter counts.
• To explore the potential of Kolmogorov-Arnold Networks (KANs) as a more efficient and interpretable alternative to MLPs in solving PDEs.
• To systematically compare MLP-based and KAN-based PINNs across multiple PDE formulations—strong, energy, inverse, and weak forms—under diverse physical and geometric conditions.
• To evaluate the performance of KINN in challenging problems such as multi-scale, singular, stress-concentrated, nonlinear hyperelastic, heterogeneous, and complex geometry domains.
• To investigate the impact of scale normalization on solution stability and convergence, particularly in large-domain PDE problems.

Proposed method

• Proposes a novel framework, Kolmogorov-Arnold-Informed Neural Network (KINN), by replacing the fully connected layers of PINNs with KANs, which use learnable univariate functions instead of fixed activation functions.
• Employs the Kolmogorov-Arnold representation theorem to decompose multivariate functions into sums of univariate functions, enabling more efficient and interpretable function approximation.
• Utilizes B-splines for constructing the learnable activation functions in KANs, with alternative variants explored using Chebyshev polynomials, radial basis functions, and wavelets.
• Applies multiple PDE forms—strong, energy (DEM), inverse (BINN), and weak (hp-VPINN)—within the KAN framework to assess computational equivalence and efficiency.
• Introduces scale normalization by rescaling spatial coordinates via $ \frac{\mathbf{x}}{L} $, where $ L $ is the characteristic size, to mitigate large gradient effects in derivatives and improve training stability.
• Employs a unified loss function combining PDE residuals, boundary conditions, and, in energy form, variational energy minimization to train the KINN model.

Experimental results

Research questions

• RQ1 Can KAN-based neural networks outperform traditional MLP-based PINNs in solving forward and inverse PDE problems in computational solid mechanics?
• RQ2 How does the use of learnable univariate activation functions in KANs affect the convergence speed and solution accuracy compared to fixed activation functions in MLPs?
• RQ3 Does the KINN framework maintain superior performance across different PDE formulations (strong, energy, inverse, weak forms), or is performance sensitive to the chosen formulation?
• RQ4 How does scale normalization influence the stability and accuracy of KINN solutions in large-domain PDE problems?
• RQ5 In what types of problems—such as multi-scale, singular, or complex geometry—does KINN show particular advantages or limitations compared to MLP-based PINNs?

Key findings

• KINN significantly outperforms MLP-based PINNs in terms of solution accuracy and convergence speed for most PDE problems, including multi-scale, singular, stress-concentrated, nonlinear hyperelastic, and heterogeneous material problems.
• The performance gain in KINN is attributed to the KAN’s learnable activation functions, which provide better function approximation with fewer parameters compared to fixed activation functions in MLPs.
• In problems involving large characteristic sizes, scale normalization is critical: without it, coordinate derivatives can grow unbounded due to the $ x \cdot \partial_x \phi $ term, leading to unstable gradients and poor convergence.
• With scale normalization, KINN maintains stable training and accurate solutions, and the solution scales linearly with the domain size, preserving physical consistency.
• KINN shows comparable performance to MLPs in complex geometry problems, suggesting that geometric complexity may limit the advantage of KANs in such cases.
• The energy form (DEM) of KINN demonstrates particular robustness, especially when combined with scale normalization, due to the variational nature of energy minimization.