Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the deep learning-based techniques. Based on the data and physical models, PINNs introduce the standard feedforward neural networks (NNs) to approximate the solutions to the PDE systems. By using automatic differentiation, the PDEs system could be explicitly encoded into NNs and consequently, the sum of mean squared residuals from PDEs could be minimized with respect to the NN parameters. In this study, the residual in the loss function could be constructed validly by using the automatic differentiation because of the relationship between the surface differential operators $\nabla_S/\Delta_S$ and the standard Euclidean differential operators $\nabla/\Delta$. We first consider the unit sphere as surface to investigate the numerical accuracy and convergence of the PINNs with different training example sizes and the depth of the NNs. Another examples are provided with different complex manifolds to verify the robustness of the PINNs.

中文翻译：

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物理信息神经网络，用于3D流形上的椭圆型偏微分方程

基于对物理信息神经网络（PINN）的最新研究的启发，我们首次尝试引入PINN，以对3D流形上的椭圆形偏微分方程（PDE）进行数值模拟。PINN是基于深度学习的技术之一。基于数据和物理模型，PINN引入了标准前馈神经网络（NNs）来近似PDE系统的解决方案。通过使用自动微分，可以将PDEs系统显式编码为NN，因此，可以相对于NN参数最小化PDE的均方差之和。在这项研究中，由于表面微分算子$ \ nabla_S / \ Delta_S $与标准欧几里德微分算子$ \ nabla / \ Delta $之间的关系，可以使用自动微分有效地构造损失函数中的残差。我们首先将单位球面作为表面，以研究具有不同训练示例大小和神经网络深度的PINN的数值准确性和收敛性。提供了带有不同复杂歧管的另一个示例，以验证PINN的鲁棒性。