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

2026-02-19

AI Summary

This work proposes a novel approach to solving partial differential equations (PDEs) by integrating the weak formulation into evolutionary Kolmogorov–Arnold Networks (KANs). Unlike strong-form methods, which often yield ill-conditioned linear systems and suffer from rapidly escalating computational costs with increasing sample size, the proposed framework leverages a variational principle to decouple the linear system dimension from the number of training samples. Furthermore, it introduces boundary-constrained KANs that rigorously embed Dirichlet, Neumann, and periodic boundary conditions directly into the network architecture. This strategy substantially enhances solution stability and scalability, achieving high accuracy while significantly reducing computational complexity. The method thus establishes an efficient new paradigm for large-scale scientific machine learning and engineering applications involving PDEs.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Search and Optimization: Evolutionary Computation

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Large pretrained models with web data

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.

Problem

• partial differential equations
• evolutionary neural networks
• strong-form formulation
• ill-conditioned linear systems
• scalability

Innovation

• weak-form formulation
• evolutionary KAN
• boundary-constrained neural networks
• scalable PDE solver
• scientific machine learning