arXiv 2606.271262026-06-26cs.LGphysics.data-anphysics.flu-dyn

Kolmogorov Arnold网络（KAN）用于气动预测：与MLP和GNN的比较

Kolmogorov Arnold networks (KAN) for aerodynamic prediction: a comparison with MLPs and GNNs

ETSIAE-UPM-School of Aeronautics, Universidad Politécnica de Madrid（马德里理工大学航空学院）；Center for Computational Simulation, Universidad Politécnica de Madrid（马德里理工大学计算模拟中心）；Institute for Cross-Disciplinary Physics and Complex Systems (IFISC, CSIC-UIB)（跨学科物理与复杂系统研究所（IFISC，CSIC-UIB））

Miguel Jaraiz, Fermin Gutierrez, Pablo Yeste, Miguel Sánchez-Domínguez, Eusebio Valero, Gonzalo Rubio, Lucas Lacasa

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AI总结：

本文比较了KAN、MLP和GNN在预测亚音速和跨音速翼型表面压力分布上的性能，发现KAN性能略逊于MLP，且训练不稳定，而GNN性能最佳但训练更耗时。

摘要：

Kolmogorov Arnold网络（KAN）最近被引入作为一种（深度）神经网络架构，其可训练参数调整激活函数，而不是传统架构（如深度多层感知器（MLP））核心的仿射变换系数。该架构基于Kolmogorov-Arnold定理，具有通用逼近性质。尽管KAN的出现令人兴奋，但目前存在关于KAN是否能在符号回归、通用机器学习、自然语言处理或计算机视觉等经典领域超越深度多层感知器（MLP）的争论。本文评估了KAN在流体动力学代理建模领域的性能，并将其与MLP和图神经网络（GNN）进行了细致比较。为此，我们考虑了预测亚音速和跨音速翼型表面压力分布的任务，这是空气动力学中的经典任务。我们的结果表明，KAN模型在预测整个压力系数方面表现良好，能够跨马赫数和攻角进行插值，但其性能与适当训练的MLP相当（略逊一筹），而最佳性能由GNN实现，但代价是需要更长的训练时间。尽管最优KAN模型的复杂度通常远低于MLP和GNN（因此训练更快），但我们发现KAN存在训练不稳定性，其性能高度依赖于适当的超参数优化。

英文摘要：

Kolmogorov Arnold networks (KAN) have recently been introduced as a (deep) neural network architecture whose trainable parameters adapt the activation functions, instead of the coefficients of the affine transformations at the core of traditional architectures such as deep multilayer perceptrons (MLPs). This architecture builds on the Kolmogorov-Arnold theorem, which endows it with universal approximation properties. While the advent of KANs has been received with excitement, there is a current debate about the possible KAN supremacy over deep multilayer perceptrons (MLPs) for classic fields such as symbolic regression, generic-purpose machine learning, natural language processing or computer vision. Here we assess the performance of KANs --and its nuanced comparison against MLPs and graph neural networks (GNNs)-- in the realm of fluid dynamics surrogate modelling. To that aim, we consider the task of predicting the surface pressure distribution over subsonic and transonic airfoils, a canonical task in aerodynamics. Our results show that KAN models show good performance in predicting the whole pressure coefficients and is able to interpolate across Mach numbers and angles of attack, however its performance is comparable --marginally inferior-- to a suitably trained MLP, where best performance is achieved by a GNN at the expense or requiring lengthier training. While the optimal KAN model have typically much lower complexity than MLP and GNN --hence resulting in faster training--, we find that KANs suffer from training instabilities, and their performance is highly dependent on a proper hyperparameter optimisation.

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