Kolmogorov–Arnold Networks (KANs) have emerged as a promising alternative to traditional Multilayer Perceptrons (MLPs) in deep learning. KANs have already been integrated into various architectures, such as convolutional neural networks, graph neural networks, and transformers, and their potential has been assessed for predicting physical quantities. However, the combination of KANs with point-cloud-based neural networks (e.g., PointNet) for computational physics has not yet been explored. To address this, we present Kolmogorov–Arnold PointNet (KA-PointNet) as a novel supervised deep learning framework for the prediction of incompressible steady-state fluid flow fields in irregular domains, where the predicted fields are a function of the geometry of the domains. In KA-PointNet, we implement shared KANs in the segmentation branch of the PointNet architecture. We utilize Jacobi polynomials to construct shared KANs. As a benchmark test case, we consider incompressible laminar steady-state flow over a cylinder, where the geometry of its cross-section varies over the data set. We investigate the performance of Jacobi polynomials with different degrees as well as special cases of Jacobi polynomials such as Legendre polynomials, Chebyshev polynomials of the first and second kinds, and Gegenbauer polynomials, in terms of the computational cost of training and accuracy of prediction of the test set. Furthermore, we examine the robustness of KA-PointNet in the presence of noisy training data and missing points in the point clouds of the test set. Additionally, we compare the performance of PointNet with shared KANs (i.e., KA-PointNet) and PointNet with shared MLPs. It is observed that when the number of trainable parameters is approximately equal, PointNet with shared KANs (i.e., KA-PointNet) outperforms PointNet with shared MLPs. Moreover, KA-PointNet predicts the pressure and velocity distributions along the surface of cylinders more accurately, resulting in more precise computations of lift and drag.

中文翻译：

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Kolmogorov–Arnold PointNet：用于预测不规则几何形状上流体场的深度学习


Kolmogorov-Arnold 网络 （KAN） 已成为深度学习中传统多层感知器 （MLP） 的有前途的替代品。KAN 已经集成到各种架构中，例如卷积神经网络、图神经网络和 Transformer，并且已经评估了它们在预测物理量方面的潜力。然而，尚未探索将 KAN 与基于点云的神经网络（例如 PointNet）相结合用于计算物理学。为了解决这个问题，我们将 Kolmogorov-Arnold PointNet （KA-PointNet） 作为一种新的监督深度学习框架，用于预测不规则域中的不可压缩稳态流体流场，其中预测场是域几何形状的函数。在 KA-PointNet 中，我们在 PointNet 架构的分段分支中实施了共享 KAN。我们利用雅可比多项式来构建共享 KAN。作为一个基准测试用例，我们考虑了圆柱体上的不可压缩层流稳态流，其中其横截面的几何形状在数据集中发生变化。我们研究了不同程度的雅可比多项式的性能，以及雅可比多项式的特殊情况，如勒让德多项式、第一类和第二种切比雪夫多项式以及格根鲍尔多项式，在训练的计算成本和测试集预测的准确性方面。此外，我们检查了 KA-PointNet 在存在嘈杂训练数据和测试集点云中缺失点时的鲁棒性。此外，我们还比较了 PointNet 与共享 KAN（即 KA-PointNet）和 PointNet 与共享 MLP 的性能。据观察，当可训练参数的数量大致相等时，具有共享 KAN 的 PointNet（即，KA-PointNet）的性能优于具有共享 MLP 的 PointNet。此外，KA-PointNet 可以更准确地预测沿圆柱体表面的压力和速度分布，从而更精确地计算升力和阻力。