Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometries

Kolmogorov-Arnold Networks (KANs) have gained attention as an 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 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. We construct shared KANs using Jacobi polynomials and investigate their performance by considering Jacobi polynomials of different degrees and types in terms of both computational cost and prediction accuracy. As a benchmark test case, we consider natural convection in a square enclosure with a cylinder, where the cylinder's shape varies across a dataset of 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 using MLPs, we observe that, with approximately the same number of trainable parameters and comparable computational cost in terms of the number of epochs, training time per epoch, and memory usage, PI-KAN-PointNet yields more accurate predictions, particularly for values on unknown boundary conditions involving nonsmooth geometries.