标题： 基于物理信息的KAN PointNet：用于不可压缩流在众多不规则几何形状上同时求解逆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）作为一种有前途的替代方案，引起了人们在计算物理领域特别是物理学信息神经网络（PINNs）框架下对深度学习应用的关注。 引入了带有其变体的物理学信息的Kolmogorov-Arnold网络（PIKANs），并对其求解反问题进行了评估。 然而，与PINNs类似，当前版本的PIKANs每次训练只能获得单个计算域的解；因此，新的几何形状需要从头开始重新训练模型。 为了克服这一限制，提出了物理学信息的PointNet（PIPN）。 在这项工作中，我们引入了物理学信息的Kolmogorov-Arnold PointNet（PI-KAN-PointNet），以扩展PIKANs的能力。 PI-KAN-PointNet能够在单一训练过程中同时解决多个不规则几何形状的反问题，从而降低成本。 我们使用雅可比多项式构建KAN，并通过考虑不同度数和类型的雅可比多项式来研究其性能，涉及计算成本和预测精度。 作为基准测试案例，我们考虑了具有圆柱体的方形封闭区域内的自然对流，其中圆柱体的形状在135个几何形状的数据集中变化。 我们将PI-KAN-PointNet的表现与PIPN（即带有MLP的物理学信息PointNet）进行比较，并观察到，在参数数量大致相等且计算成本相似的情况下，PI-KAN-PointNet提供了更准确的预测。 最后，我们探索了在构建物理学信息PointNet时结合KAN和MLP。 我们的研究结果表明，采用MLP层作为编码器和KAN层作为解码器的物理学信息PointNet模型在所有研究的模型中是最优配置。 显示英文摘要

摘要： Kolmogorov-Arnold Networks (KANs) have gained attention as a promising alternative to traditional Multilayer Perceptrons (MLPs) for deep learning applications in computational physics, especially within the framework of physics-informed neural networks (PINNs). Physics-informed Kolmogorov-Arnold Networks (PIKANs) and their variants have been introduced and evaluated to solve inverse problems. However, similar to PINNs, current versions of PIKANs are limited to obtaining solutions for a single computational domain per training run; consequently, a new geometry requires retraining the model from scratch. Physics-informed PointNet (PIPN) was introduced to address this limitation for PINNs. In this work, we introduce physics-informed Kolmogorov-Arnold PointNet (PI-KAN-PointNet) to extend this capability to PIKANs. PI-KAN-PointNet enables the simultaneous solution of an inverse problem over multiple irregular geometries within a single training run, reducing computational costs. We construct 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. We compare the performance of PI-KAN-PointNet with that of PIPN (i.e., physics-informed PointNet with MLPs) and observe that, with approximately an equal number of trainable parameters and similar computational cost, PI-KAN-PointNet provides more accurate predictions. Finally, we explore the combination of KAN and MLP in constructing a physics-informed PointNet. Our findings indicate that a physics-informed PointNet model employing MLP layers as the encoder and KAN layers as the decoder represents the optimal configuration among all models investigated.