用于粘弹性流体方程的物理信息型柯尔莫哥洛夫-阿诺尔德网络

中文摘要 AI 辅助

柯尔莫哥洛夫-阿诺尔德网络（KANs）受柯尔莫哥洛夫-阿诺尔德表示定理启发，通过在边使用可学习激活函数而非固定节点激活函数，提供了多层感知机（MLPs）的可解释替代方案。我们提出一种物理信息型柯尔莫哥洛夫-阿诺尔德网络（PI-KAN）框架，用于求解粘弹性流体方程的正问题，这类方程出现在许多复杂流体动力学应用中，具有流体场间强非线性耦合的特征。对于粘弹性流体方程，我们采用在尘埃等离子体领域已成熟的广义流体动力学模型。为评估所提框架在粘弹性流体问题上的性能，我们考虑基于泰勒-格林（TG）流及改进型泰勒-格林流的基准问题，系统研究不同网络架构、超参数及配点分布对τₘ=1--20范围内PI-KAN精度与收敛行为的影响，还分析随机种子初始化对训练结果的影响。所得结果为设计和实现用于求解粘弹性流体方程的物理信息型柯尔莫哥洛夫-阿诺尔德网络（PI-KAN）提供了有用指导。

英文摘要

Kolmogorov-Arnold Networks (KANs), inspired by the Kolmogorov Arnold representation theorem, provide an interpretable alternative to multilayer perceptrons (MLPs) by using learnable activation functions on edges rather than fixed node activations. We propose a Physics-Informed Kolmogorov-Arnold Network (PI-KAN) framework for solving forward problem of viscoelastic fluid equations, which arise in many complex fluid dynamics applications and are characterized by strong nonlinear coupling between fluid fields. For viscoelastic fluid equations, we adopt the generalized hydrodynamic model, which is well established in the field of dusty plasma. To evaluate the performance of the proposed framework for viscoelastic fluid, we consider benchmark problem based on the Taylor-Green (TG) flow and a modified Taylor-Green flow. We systematically investigate the effects of different network architectures, hyperparameters, and collocation point distributions on the accuracy and convergence behavior of PI-KANs for the range of viscoelastic parameter ($τ_m = 1$--$20$). We also study the impact of random seed initialization on training outcomes. The obtained results provide useful guidance for the design and implementation of physics-informed Kolmogorov-Arnold networks (PI-KANs) in solving viscoelastic fluid equations

发表机构

• Indian Institute of Science Education and Research (IISER), Berhampur（印度科学教育与研究学院（IISER），贝尔汉普尔）
• Maulana Azad National Institute of Technology (MANIT)（莫拉纳·阿扎德国立理工学院（MANIT））

机构由 AI 辅助整理，请以论文原文为准。

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17 pages, 15 figures