  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
Kolmogorov-Arnold Networks (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
Multilayer Perceptrons (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.
