物理信息 PointNet：它能在多少不规则几何体上同时求解一个反问题？在线性弹性中的应用
Physics-informed PointNet: On how many irregular geometries can it solve an inverse problem simultaneously? Application to linear elasticity
AI总结：
本文提出物理信息 PointNet（PIPN），在仅用稀疏标注数据的情况下同时预测 500 多个不同几何域上的平面应力问题解，并系统研究批大小、网络规模、对称函数及损失权重的影响。
摘要：
常规的物理信息神经网络（PINNs）利用稀疏标注数据预测偏微分方程的解，但仅限于单一域。另一方面，全监督学习模型通常先在数千个具有已知解（即标注数据）的域上进行训练，然后在数百个未见过的域上预测解。物理信息 PointNet（PIPN）主要旨在填补 PINNs（作为弱监督学习模型）与全监督学习模型之间的这一空白。在本文中，我们证明 PIPN 能够在仅使用稀疏标注数据的情况下，同时在数百个域上预测所需偏微分方程的解。该框架有利于工业中仅有稀疏标注数据可用时的快速几何设计。特别地，我们表明 PIPN 能够同时在 500 多个具有不同几何形状的域上预测平面应力问题的解。此外，我们率先在 PIPN 中实现了显著批大小（即每个子 epoch 输入 PIPN 的几何体数量）的概念。具体而言，我们尝试了 7、14、19、38、76 和 133 的批大小。另外，还研究了 PIPN 规模、PIPN 架构中的对称函数以及损失函数中稀疏标注数据分量的静态和动态权重的影响。
英文摘要：
Regular physics-informed neural networks (PINNs) predict the solution of partial differential equations using sparse labeled data but only over a single domain. On the other hand, fully supervised learning models are first trained usually over a few thousand domains with known solutions (i.e., labeled data) and then predict the solution over a few hundred unseen domains. Physics-informed PointNet (PIPN) is primarily designed to fill this gap between PINNs (as weakly supervised learning models) and fully supervised learning models. In this article, we demonstrate that PIPN predicts the solution of desired partial differential equations over a few hundred domains simultaneously, while it only uses sparse labeled data. This framework benefits fast geometric designs in the industry when only sparse labeled data are available. Particularly, we show that PIPN predicts the solution of a plane stress problem over more than 500 domains with different geometries, simultaneously. Moreover, we pioneer implementing the concept of remarkable batch size (i.e., the number of geometries fed into PIPN at each sub-epoch) into PIPN. Specifically, we try batch sizes of 7, 14, 19, 38, 76, and 133. Additionally, the effect of the PIPN size, symmetric function in the PIPN architecture, and static and dynamic weights for the component of the sparse labeled data in the loss function are investigated.
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