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[2602.18515] Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations
Summary
This article presents a novel weak-form evolutionary Kolmogorov-Arnold Network (KAN) designed to solve partial differential equations (PDEs) more efficiently, addressing limitations of strong-form approaches in scientific computing.
Why It Matters
The proposed KAN framework enhances the scalability and accuracy of PDE solutions, which is crucial for advancements in scientific machine learning and engineering applications. By decoupling linear system size from training samples, it offers a promising solution to computational challenges in this field.
Key Takeaways
- Weak-form evolutionary KANs improve scalability for solving PDEs.
- The approach rigorously enforces boundary conditions for accuracy.
- Decoupling linear system size from training samples reduces computational costs.
- This method contributes to the field of scientific machine learning.
- Potential applications include various engineering disciplines.
Computer Science > Machine Learning arXiv:2602.18515 (cs) [Submitted on 19 Feb 2026] Title:Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations Authors:Bongseok Kim, Jiahao Zhang, Guang Lin View a PDF of the paper titled Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations, by Bongseok Kim and 2 other authors View PDF 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 conditi...
