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

TL;DR

This paper tackles the challenge of enforcing boundary conditions in data-driven PDE solvers by proposing BEKAN, a boundary condition–guaranteed evolutionary Kolmogorov–Arnold network with Gaussian radial basis functions. BEKAN integrates three mechanisms: Dirichlet conditions embedded directly into the KAN basis, a periodic layer that guarantees exact periodicity, and an evolutionary update driven by a least-squares formulation to satisfy Neumann BCs over time. Across five benchmark PDEs (Dirichlet, Neumann, periodic, and mixed BCs), BEKAN achieves higher accuracy and exact boundary satisfaction compared to MLP-based approaches and B-splines KAN, while maintaining stable training and well-conditioned Jacobians, even for chaotic or stiff problems like KS. This framework facilitates reliable, boundary-faithful PDE simulations with potential extensions to uncertainty quantification and broader scientific computing applications.

Abstract

Deep learning has gained attention for solving PDEs, but the black-box nature of neural networks hinders precise enforcement of boundary conditions. To address this, we propose a boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (KAN) with radial basis functions (BEKAN). In BEKAN, we propose three distinct and combinable approaches for incorporating Dirichlet, periodic, and Neumann boundary conditions into the network. For Dirichlet problem, we use smooth and global Gaussian RBFs to construct univariate basis functions for approximating the solution and to encode boundary information at the activation level of the network. To handle periodic problems, we employ a periodic layer constructed from a set of sinusoidal functions to enforce the boundary conditions exactly. For a Neumann problem, we devise a least-squares formulation to guide the parameter evolution toward satisfying the Neumann condition. By virtue of the boundary-embedded RBFs, the periodic layer, and the evolutionary framework, we can perform accurate PDE simulations while rigorously enforcing boundary conditions. For demonstration, we conducted extensive numerical experiments on Dirichlet, Neumann, periodic, and mixed boundary value problems. The results indicate that BEKAN outperforms both multilayer perceptron (MLP) and B-splines KAN in terms of accuracy. In conclusion, the proposed approach enhances the capability of KANs in solving PDE problems while satisfying boundary conditions, thereby facilitating advancements in scientific computing and engineering applications.

Figures (22)

• Figure 1: Evolutionary Kolmogorov-Arnold networks with Gaussian RBFs.
• Figure 2: Boundary condition-guaranteed evolutionary KAN with radial basis functions: Dirichlet boundary condition. The illustrated $[2, 3, 3, 1]$ architecture depicted the use of $h_1$- and $h_2$-scaled activation layers, where $h_1$ ensures vanishing at the domain boundaries and $h_2$ maps zero inputs to zero outputs to enforce homogeneous conditions. For non-homogeneous boundary condition, a lifting function $l(x,t)$ is employed.
• Figure 3: Boundary condition-guaranteed evolutionary KAN with radial basis functions: Periodic boundary condition.
• Figure 4: Boundary condition-guaranteed evolutionary KAN with radial basis functions: Neumann boundary condition.
• Figure 5: Evolution of the original energy $E$ and the modified energy $r^2$ during the training process of BEKAN for the 1D Allen–Cahn equation (Eq. \ref{['eq:Allen-Cahn']}). Each iteration corresponds to one forward step in the time integration with a time increment $\Delta t = 1.0 \times 10^{-6}$. The formulation is constructed such that the modified energy $r^2$ converges to the original energy $E$, thereby ensuring stable energy evolution throughout the training.

Theorems & Definitions (1)

• Remark 3.1