Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks (2406.11045v2)

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.

• The paper introduces the KINN framework, integrating Kolmogorov-Arnold Networks into PDE-solving for enhanced accuracy and faster convergence.
• It employs learned activation functions based on B-splines to overcome spectral bias and improve performance on multi-scale, nonlinear, and heterogeneous problems.
• Numerical experiments demonstrate significant improvements over traditional PINNs and FEM, highlighting KINN’s potential in computational physics applications.

Overview of the Kolmogorov–Arnold-Informed Neural Network (KINN)

The paper under discussion introduces the Kolmogorov–Arnold-Informed Neural Network (KINN), a novel framework designed to improve the accuracy and efficiency of solving partial differential equations (PDEs) using deep learning techniques. Building on the principles of Kolmogorov-Arnold representation, the authors propose leveraging Kolmogorov–Arnold Networks (KANs) to augment the existing Physics-Informed Neural Networks (PINNs). The intent is to harness KAN's interpretability and parameter efficiency to address various forms of PDEs including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous materials, and complex geometries.

Core Contributions and Methodology

The cornerstone of this work is the introduction of KAN into the field of PDE-solving frameworks, specifically within three different formulations of PDEs: the strong form (PINNs), the energy form (Deep Energy Method - DEM), and the inverse form (Boundary-Integral Neural Networks - BINNs). The research provides a rigorous comparison between the Multi-Layer Perceptron (MLP) based PINNs and their KAN-based counterparts (KINN).

The paper is organized as follows:

1. Introduction and Background:
   • A concise overview of the mathematical equivalence yet computational differences of various PDE forms.
   • Discussion on the drawbacks of traditional MLPs, including spectral bias issues and lack of interpretability.
2. KAN Architecture and KINN Framework:
   • Detailed exploration of KAN, emphasizing the benefits of learned activation functions, specifically constructed via B-splines.
   • The integration of KAN into different forms of PDEs (PINNs, DEM, and BINN), optimizing their respective loss functions.
3. Numerical Experiments:
   • Extensive validation of KINN within benchmark problems, highlighting significant improvements in accuracy and convergence speed compared to MLP-based models, except for scenarios with complex geometries.
   • Performance metrics across various examples such as multi-scale, singularity, stress concentration, nonlinear hyperelasticity, and heterogeneous materials are systematically reported.

Strong Numerical Results and Notable Findings

The numerical results presented in the paper demonstrate the robustness and advantages of KINN across different classes of PDE problems. Specifically:

• Improved Accuracy and Convergence: KINN outperforms traditional MLP-based models, providing notable improvements in both convergence speed and accuracy. For example, in handling multi-scale problems where traditional PINNs struggle due to spectral bias, KINN shows substantial efficacy by maintaining robust performance across both high and low-frequency components.
• Heterogeneous Problems: KAN's interpretability and parameter efficiency allow for more accurate solutions of heterogeneous material problems without the need for complex domain decompositions as required by CPINNs or CENN.
• Comparison with FEM and Traditional Methods: KINN also illustrates competitive or superior accuracy when benchmarked against conventional numerical methods such as Finite Element Methods (FEM).

Theoretical and Practical Implications

The paper posits significant practical and theoretical implications:

• Practical Utility: With its application to real-world problems, KINN could substantially improve computational mechanics, making it a valuable tool for engineers and scientists dealing with complex systems modeled by PDEs.
• Future Potential: The methodology opens avenues for embracing neural networks that adhere closely to traditional numerical algorithms, ultimately enhancing the interpretability and efficiency of AI-driven scientific computation.

Limitations and Future Research Directions

Despite its promising results, KINN faces challenges in handling complex geometric domains, as KAN's performance diminishes. Future research might focus on refining grid size adaptability and leveraging mesh adaptation techniques from FEM, such as h-p refinements or isoparametric transformation methods, to enhance KAN’s capabilities in complex geometries.

Additionally, exploring weak form PDEs and other advanced integration schemes could further broaden the applicability and strength of KINN. Extending the framework to data-driven inverse problems could also reveal new utilities of KAN in discovering symbolic representations of complex functions from empirical data.

Conclusion

The Kolmogorov–Arnold-Informed Neural Network framework proposed in this paper establishes a compelling enhancement over traditional MLP-based PINNs for solving a variety of PDEs. By leveraging KAN’s interpretability and parameter efficiency, the paper marks a significant step towards more accurate and efficient AI-driven solutions in computational physics. While challenges remain, particularly in complex geometries, the framework sets a robust foundation for future advancements in AI for PDEs.