Physics‐Informed KAN PointNet: Deep Learning for Simultaneous Solutions to Inverse Problems in Incompressible Flow on Numerous Irregular Geometries

ABSTRACT Kolmogorov–Arnold Networks (KANs) have gained attention as a promising alternative to traditional multilayer perceptrons (MLPs) for deep learning applications in computational physics, particularly for solving inverse problems with sparse data, as exemplified by the physics‐informed Kolmogorov‐Arnold network (PIKAN). However, the capability of KANs to simultaneously solve inverse problems over multiple irregular geometries within a single training run remains unexplored. To address this gap, we introduce the physics‐informed Kolmogorov‐Arnold PointNet (PI‐KAN‐PointNet), in which shared KANs are integrated into the PointNet architecture to capture the geometric features of the computational domains. The loss function comprises the squared residuals of the governing equations, computed via automatic differentiation, along with sparse observations and partially known boundary conditions. Jacobi polynomials are used in the shared KANs. A comparison between various degrees and types of the polynomials is made from the perspectives of computational cost and prediction accuracy. As a case study, we investigate the free convection of a fluid flowing between a square cylinder and another cylinder with varying shapes for 135 geometries. PI‐KAN‐PointNet offers two main advantages. First, it overcomes the limitation of current PIKANs, which are restricted to solving only a single computational domain per training run, thereby reducing computational costs. Second, when comparing the performance of PI‐KAN‐PointNet with that of the physics‐informed PointNet (PIPN) using MLPs, we observe that, with approximately the same number of trainable parameters and comparable computational costs in terms of the number of epochs, training time per epoch, and memory usage, PI‐KAN‐PointNet yields more accurate predictions, particularly for values of unknown boundary conditions involving non‐smooth geometries.