标题： 基于物理信息的KAN PointNet：在多个不规则几何体上不可压缩流动反问题的深度学习方法 显示英文标题

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

摘要： Kolmogorov-Arnold网络（KANs）因其在计算物理领域的深度学习应用，特别是解决稀疏数据的反问题，而成为传统多层感知机（MLPs）的一种替代方案，如物理信息Kolmogorov-Arnold网络（PIKAN）所示。 然而，KANs在单次训练运行中同时解决多个不规则几何的反问题的能力仍未经探索。 为解决这一空白，我们引入了物理信息Kolmogorov-Arnold PointNet（PI-KAN-PointNet），其中共享的KANs被整合到PointNet架构中，以捕捉计算域的几何特征。 损失函数包括通过自动微分计算的控制方程的平方残差，以及稀疏观测和部分已知的边界条件。 我们使用雅可比多项式构建共享的KANs，并通过考虑不同次数和类型的雅可比多项式来研究其性能，从计算成本和预测准确性方面进行评估。 作为基准测试案例，我们考虑了一个带有圆柱体的正方形腔体中的自然对流问题，其中圆柱体的形状在135个几何数据集中变化。 PI-KAN-PointNet具有两个主要优势。 首先，它克服了当前PIKANs的限制，这些限制仅能每轮训练解决一个计算域，从而降低了计算成本。 其次，当将PI-KAN-PointNet与使用MLPs的物理信息PointNet的性能进行比较时，我们发现，在可训练参数数量大致相同且在epoch数量、每个epoch的训练时间和内存使用方面具有可比计算成本的情况下，PI-KAN-PointNet的预测更加准确，尤其是在涉及非光滑几何的未知边界条件值方面。 显示英文摘要

摘要： 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.