Enhancing Physics-Informed Neural Networks with a Hybrid Parallel Kolmogorov-Arnold and MLP Architecture

TL;DR

This paper addresses training efficiency and accuracy challenges in physics-informed neural networks by introducing HPKM-PINN, a hybrid parallel architecture that fuse two distinct networks, a Kolmogorov-Arnold Network (KAN) and an MLP, through a tunable weight $ξ$. The approach combines a standard PINN loss with parallel branches whose outputs are blended as $u(x)=ξ\,u_{KAN}(x)+(1-ξ)\,u_{MLP}(x)$, enabling robust learning of both high- and low-frequency features. Across function approximation and four PDE benchmarks (Poisson, Advection, Convection–Diffusion, Helmholtz), HPKM-PINN achieves significantly lower losses and faster convergence than standalone PINN or KAN variants, and demonstrates robustness to noise with task-specific optimal mixings. These results underscore the value of hybrid, parallel architectures for accurate, scalable PDE solving in computational science and engineering, while highlighting trade-offs in model size and training cost. The work suggests future directions in adaptive balancing and optimization to further enhance efficiency and applicability.

Abstract

Neural networks have emerged as powerful tools for modeling complex physical systems, yet balancing high accuracy with computational efficiency remains a critical challenge in their convergence behavior. In this work, we propose the Hybrid Parallel Kolmogorov-Arnold Network (KAN) and Multi-Layer Perceptron (MLP) Physics-Informed Neural Network (HPKM-PINN), a novel architecture that synergistically integrates parallelized KAN and MLP branches within a unified PINN framework. The HPKM-PINN introduces a scaling factor ξ, to optimally balance the complementary strengths of KAN's interpretable function approximation and MLP's nonlinear feature learning, thereby enhancing predictive performance through a weighted fusion of their outputs. Through systematic numerical evaluations, we elucidate the impact of the scaling factor ξ on the model's performance in both function approximation and partial differential equation (PDE) solving tasks. Benchmark experiments across canonical PDEs, such as the Poisson and Advection equations, demonstrate that HPKM-PINN achieves a marked decrease in loss values (reducing relative error by two orders of magnitude) compared to standalone KAN or MLP models. Furthermore, the framework exhibits numerical stability and robustness when applied to various physical systems. These findings highlight the HPKM-PINN's ability to leverage KAN's interpretability and MLP's expressivity, positioning it as a versatile and scalable tool for solving complex PDE-driven problems in computational science and engineering.

Figures (13)

• Figure 1: Hybrid Parallel KAN-MLP PINN architecture. On the left side of the figure is the neural network component, while the right side represents the physics-informed component. The neural network part receives two input variables, the spatial variable $x$ and the temporal variable $t$, which are simultaneously fed into two parallel networks: MLP and KAN. Each network processes the inputs through their respective hidden layers, generating outputs $Y_{1}$ and $Y_{2}$. These outputs are then combined using an adjustable network weight factor $\xi$, resulting in the final network output $u=\xi\cdot Y_2+(1-\xi)\cdot Y_1$. This output is subsequently passed to the following physics-informed component to generate the corresponding loss function term.
• Figure 2: Comparison of the Fourier spectra of the reference function, and the approximated functions obtained using (a) KAN Ratio $\xi$ = 0 (MLP), (b) KAN Ratio $\xi$ = 0.9 (hybrid structure), and (c) KAN Ratio $\xi$ = 1 (KAN).
• Figure 3: Comparison of the predicted solutions and reference solutions for Equation (14) using (a) KAN Ratio $\xi$ = 0 (MLP), (b) KAN Ratio $\xi$ = 0.9 (hybrid structure), and (c) KAN Ratio $\xi$ = 1(KAN). (d) Relationship between the $L_{2}$ error and the network weight factor.
• Figure 4: (a) Training progression of MSE loss across epochs for different KAN ratios. (b) Final convergence loss values versus KAN ratio, with error bars reflecting stability during the final 500 training epochs.
• Figure 5: Three PINN models for solving the Poisson equation. (a) MLP-based PINN, with a prediction error of 1.84%. (b) The proposed HPKM-PINN with a network weight factor $\xi$ = 0.3, yielding a prediction error of 0.29%. (c) KAN-based PIKAN, with a prediction error of 2.31%. (d) Relationship between $L_{2}$ error and network weight factors.