Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations

TL;DR

This paper introduces a weak-form evolutionary neural network approach for solving PDEs that improves scalability and stability over traditional methods by decoupling system size from training samples and rigorously enforcing boundary conditions.

Contribution

The paper proposes a novel weak-form evolutionary Kolmogorov-Arnold Network (KAN) that enhances scalability and boundary condition enforcement in PDE solutions.

Findings

Decouples linear system size from training samples

Improves scalability over strong-form methods

Enforces boundary conditions rigorously

Abstract

Partial differential equations (PDEs) form a central component of scientific computing. Among recent advances in deep learning, evolutionary neural networks have been developed to successively capture the temporal dynamics of time-dependent PDEs via parameter evolution. The parameter updates are obtained by solving a linear system derived from the governing equation residuals at each time step. However, strong-form evolutionary approaches can yield ill-conditioned linear systems due to pointwise residual discretization, and their computational cost scales unfavorably with the number of training samples. To address these limitations, we propose a weak-form evolutionary Kolmogorov-Arnold Network (KAN) for the scalable and accurate prediction of PDE solutions. We decouple the linear system size from the number of training samples through the weak formulation, leading to improved scalability compared to strong-form approaches. We also rigorously enforce boundary conditions by constructing the trial space with boundary-constrained KANs to satisfy Dirichlet and periodic conditions, and by incorporating derivative boundary conditions directly into the weak formulation for Neumann conditions. In conclusion, the proposed weak-form evolutionary KAN framework provides a stable and scalable approach for PDEs and contributes to scientific machine learning with potential relevance to future engineering applications.