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Kolmogorov-Arnold PointNet: Deep learning for prediction of fluid fields on irregular geometries

Ali Kashefi

TL;DR

KA-PointNet combines Kolmogorov-Arnold Networks with PointNet to predict incompressible flow fields on irregular 2D geometries. By replacing shared MLPs with Jacobi-polynomial-based KANs in the PointNet segmentation path, the model learns geometry-dependent activation behavior and achieves high accuracy on unseen cylinder cross-sections. The study demonstrates that Jacobi polynomial choice and degree, as well as the global scaling parameter $n_s$, critically balance predictive accuracy and computational cost, with Chebyshev variants and moderate degrees providing strong performance. KA-PointNet outperforms standard PointNet with shared MLPs in most tests, including robust drag and lift estimations, albeit at higher training cost, suggesting its potential for rapid, geometry-aware CFD predictions and future physics-informed or 3D extensions.

Abstract

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. 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.

Kolmogorov-Arnold PointNet: Deep learning for prediction of fluid fields on irregular geometries

TL;DR

KA-PointNet combines Kolmogorov-Arnold Networks with PointNet to predict incompressible flow fields on irregular 2D geometries. By replacing shared MLPs with Jacobi-polynomial-based KANs in the PointNet segmentation path, the model learns geometry-dependent activation behavior and achieves high accuracy on unseen cylinder cross-sections. The study demonstrates that Jacobi polynomial choice and degree, as well as the global scaling parameter , critically balance predictive accuracy and computational cost, with Chebyshev variants and moderate degrees providing strong performance. KA-PointNet outperforms standard PointNet with shared MLPs in most tests, including robust drag and lift estimations, albeit at higher training cost, suggesting its potential for rapid, geometry-aware CFD predictions and future physics-informed or 3D extensions.

Abstract

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. 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.
Paper Structure (21 sections, 32 equations, 21 figures, 10 tables)

This paper contains 21 sections, 32 equations, 21 figures, 10 tables.

Table of Contents

  1. Introduction and motivation
  2. Problem formulation
  3. Governing equations of interest
  4. Data generation
  5. Kolmogorov-Arnold PointNet
  6. Kolmogorov-Arnold network layers with Jacobi polynomials
  7. Architecture
  8. Shared Kolmogorov-Arnold networks
  9. Training
  10. Results and discussion
  11. General analysis
  12. Visual comparison and overall outcomes
  13. Effect of Jacobi polynomial degree
  14. Effect of Jacobi polynomial type
  15. Effect of network size
  16. ...and 6 more sections

Figures (21)

  • Figure 1: Finite volume meshes used for the numerical simulation of flow over a cylinder with a triangular cross section, with 2775 vertices on the left panel, and an elliptical cross section, with 2672 vertices on the right panel.
  • Figure 2: Examples of input and output of the generated dataset in the form of point clouds
  • Figure 3: Architecture of the Kolmogorov-Arnold PointNet. Shared KANs with the labels $(\mathcal{B}_1, \mathcal{B}_2)$ and $(\mathcal{B}_1, \mathcal{B}_2, \mathcal{B}_3)$ are explained in the text. $n_{\text{CFD}}$ denotes the number of CFD variables. $N$ is the number of points in the point clouds. $B$ represents the batch size. $n_s$ is the global scaling parameter used to control the network size. Note that the velocity and pressure fields shown are schematic.
  • Figure 4: Histograms of the relative pointwise error in $L^2$ norm for the velocity and pressure fields predicted by Kolmogorov-Arnold PointNet (i.e., KA-PointNet). The Jacobi polynomial used has a degree of 5, with $\alpha = \beta = 1$. Here, $n_s=1$ is set.
  • Figure 5: A comparison between the ground truth and prediction of the Kolmogorov-Arnold PointNet for the velocity and pressure fields after 10, 100, and 1000 epochs. The Jacobian polynomial used has a degree of 5, with $\alpha=\beta=1$. Here, $n_s=1$ is set.
  • ...and 16 more figures