BEKAN: Boundary condition–guaranteed evolutionary Kolmogorov–Arnold networks with radial basis functions for solving PDE problems

Deep learning has gained attention for solving partial differential equations (PDEs), but the limited interpretability of neural networks hinders precise enforcement of boundary conditions. To address this challenge, we propose BEKAN, a boundary condition-guaranteed evolutionary Kolmogorov-Arnold network with radial basis functions. BEKAN introduces three distinct and combinable approaches for rigorously incorporating Dirichlet, periodic, and Neumann boundary conditions.For Dirichlet problems, we construct boundary-embedded univariate basis functions using smooth Gaussian RBFs, encoding constraints directly at the activation level. For periodic problems, we design a sinusoidal periodic layer that exactly enforces periodicity. For Neumann problems, we formulate a least-squares evolutionary update that drives network parameters toward satisfying derivative constraints on boundaries. BEKAN also extends to general mixed boundary conditions, since the enforcement mechanisms for the three boundary types operate independently and can be assigned to distinct boundary segments. By integrating boundary-embedded RBFs, periodic layers, and evolutionary dynamics, BEKAN achieves accurate PDE simulations with rigorous boundary enforcement. Extensive numerical experiments on Dirichlet, Neumann, periodic, and mixed boundary value problems demonstrate that BEKAN consistently outperforms multilayer perceptron (MLP) and B-spline KAN baselines in both accuracy and boundary-condition satisfaction. This framework extends KAN capabilities for scientific machine learning, offering a robust and interpretable approach for solving complex PDEs under strict physical constraints.