arXiv:2503.23289cs.LGcs.DC2025-03被引 5

混合KAN与MLP结构,显著提升物理信息神经网络的精度与稳定性。

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

  • 并行融合KAN与MLP,用缩放因子ξ动态平衡两者优势。
  • 在泊松方程等任务中,相对误差降低两个数量级。
  • 适合需要高精度与可解释性的科学计算场景。

神经网络已成为建模复杂物理系统的重要工具,但如何在保证高精度的同时实现计算高效仍是关键挑战。本文提出混合并行柯尔莫哥洛夫-阿诺德网络与多层感知机的物理信息神经网络(HPKM-PINN),将并行的KAN与MLP分支集成于统一的PINN框架中。通过引入缩放因子ξ,优化KAN的可解释性函数逼近与MLP的非线性特征学习能力,实现输出加权融合。系统数值实验揭示了ξ对函数逼近与偏微分方程(PDE)求解性能的影响。在泊松方程、输运方程等经典PDE上,相较于独立的KAN或MLP模型,HPKM-PINN的损失值显著下降,相对误差减少两个数量级。该框架在多种物理系统中表现出良好的数值稳定性和鲁棒性。结果表明,HPKM-PINN能有效结合KAN的可解释性与MLP的表达能力,为计算科学与工程中的复杂PDE问题提供一种通用且可扩展的解决方案。

原文摘要 · Abstract (English)

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.

物理信息神经网络偏微分方程可解释性混合架构

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