The modified Physics-Informed Hybrid Parallel Kolmogorov--Arnold and Multilayer Perceptron Architecture with domain decomposition

TL;DR

This work tackles the difficulty of solving high-frequency multiscale PDEs with Physics-Informed Neural Networks by introducing a modified HPKM-PINN that blends a Kolmogorov–Arnold Network (KAN) with a Multilayer Perceptron (MLP) via a trainable, bounded weighting function S(α). It couples this architecture with overlapping domain decomposition to enable parallel subdomain solving and to mitigate global optimization challenges, supplemented by hard-constraint embeddings for boundary and initial conditions. Across Helmholtz, Poisson (2D and 5D), Reaction–Diffusion, and Allen–Cahn problems, the approach yields higher accuracy and faster convergence than single-branch MLP or KAN PINNs and demonstrates robust handling of both low- and high-frequency features. The results underscore the method’s potential for efficient, scalable AI-assisted PDE solvers, with opportunities for integration with Fourier feature embeddings and multi-level domain strategies to further enhance performance.

Abstract

In this work, we propose a modified Hybrid Parallel Kolmogorov--Arnold Network and Multilayer Perceptron Physics-Informed Neural Network to overcome the high-frequency and multiscale challenges inherent in Physics-Informed Neural Networks. This proposed model features a trainable weighting parameter to optimize the convex combination of outputs from the Kolmogorov--Arnold Network and the Multilayer Perceptron, thus maximizing the networks' capabilities to capture different frequency components. Furthermore, we adopt an overlapping domain decomposition technique to decompose complex problems into subproblems, which alleviates the challenge of global optimization. Benchmark results demonstrate that our method reduces training costs and improves computational efficiency compared with manual hyperparameter tuning in solving high-frequency multiscale problems.

Figures (30)

• Figure 1: Modified HPKM architecture.
• Figure 2: Domain decomposition of the 1D domain $\Omega = [0,l], l=1$.
• Figure 3: Graphical abstract of the modified HPKM-PINN with overlapping domain decomposition.
• Figure 4:
• Figure 5: