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.

翻译：常规的物理信息神经网络（PINNs）利用稀疏标记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利用已知解的数千个域（即标记数据）进行训练，然后预测数百个未知域的解。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空白。本文证明，PIPN可同时预测数百个域上所需偏微分方程的解，而仅需使用稀疏标记数据。该框架在工业中仅有稀疏标记数据可用时，有助于快速几何设计。特别地，我们展示了PIPN可同时预测500多个不同几何体平面应力问题的解。此外，我们开创性地将显著批量大小（即每子周期输入PIPN的几何体数量）概念引入PIPN，尝试了批量大小7、14、19、38、76和133。同时，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记数据分量的静态与动态权重的影响。