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，但仅限于单个域。而全监督学习模型通常先在数千个已知解（即标注数据）的域上进行训练，然后预测数百个未见域的解。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这一空白。本文证明，PIPN能够仅使用稀疏标注数据，同时预测数百个域上的所需偏微分方程的解。该框架在工业中仅能获取稀疏标注数据时，有利于快速几何设计。特别地，我们展示了PIPN能同时预测超过500个不同几何形状域上的平面应力问题的解。此外，我们开创性地将显著批大小（即每次子周期输入PIPN的几何形状数量）的概念引入PIPN。具体而言，我们尝试了批大小为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注数据分量的静态与动态权重的影响。