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

TL;DR

The proposed weak-form evolutionary Kolmogorov-Arnold Network framework provides a stable and scalable approach for PDEs and contributes to scientific machine learning with potential relevance to future engineering applications.

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.

Figures (16)

• Figure 1: Comparison between the Kolmogorov--Arnold Network (KAN) and a conventional fully connected DNN. (a) The KAN architecture employs functional mappings on each edge, enabling richer nonlinear representations and adaptive feature extraction. (b) The DNN architecture is composed of fully connected layers with activation functions.
• Figure 2: Evolutionary Kolmogorov--Arnold network with Gaussian RBFs: Strong-form formulation. At each time step, the parameters ${W}(t)$ evolve by minimizing the PDE residual functional $\mathcal{J}(\gamma)=\tfrac{1}{2}\!\int_{\Omega}\!\|\tfrac{\partial \hat{u}}{\partial {W}}\gamma+\mathcal{N}(\hat{u})\|_2^2\,\mathrm{d}\mathbf{x}$, where the optimal update direction satisfies $\frac{\partial {W}}{\partial t}=\arg\min_{\gamma}\mathcal{J}(\gamma)$. After obtaining $\gamma_{\text{opt}}$, the network parameters are explicitly updated in time using the forward Euler scheme ${W}(t_{n+1}) = {W}(t_n) + \gamma_{\text{opt}}\Delta t$, yielding a discrete-time parameter evolution associated with the strong-form PDE residual minimization.
• Figure 3: Weak-form evolutionary Kolmogorov--Arnold network. At each time step, the parameters $W(t)$ are updated by minimizing the weak--form residual functional $\mathcal{J}(\gamma) = \tfrac{1}{2}\sum_{k=1}^{K}\!\left(\int_{\Omega}(\tfrac{\partial \hat{u}}{\partial W}\gamma + \mathcal{N}(\hat{u}))v_k(\mathbf{x})\,\mathrm{d}\mathbf{x}\right)^{2}$, where $\{v_k\}_{k=1}^{K}$ denote the test functions. The optimal update direction is given by $\gamma_{\text{opt}} = \arg\min_{\gamma}\mathcal{J}(\gamma)$, and the network parameters are advanced in time via the forward Euler step $W(t_{n+1}) = W(t_n) + \gamma_{\text{opt}}\Delta t$, resulting in a discrete-time parameter evolution associated with the weak-form residual minimization.
• Figure 4: Comparison of predicted solutions for the one-dimensional Allen--Cahn equation (Eq. \ref{['eq:AC_simple_eps']}) at $t = 2 \times 10^{-5}$, trained with 100 data points. Both EvoKAN--WF and EvoKAN--SF closely match the ground-truth solution at this time, whereas the vanilla PINN does not reproduce the correct profile.
• Figure 5: Comparison of predicted solutions for the one-dimensional Allen--Cahn equation (Eq. \ref{['eq:AC_simple_eps']}) at $t = 5 \times 10^{-5}$, trained with 100 data points. As the solution profile becomes steeper, EvoKAN--WF maintains good accuracy, whereas EvoKAN--SF shows small deviations near the center and boundaries, and the vanilla PINN fails to recover the correct solution.