  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.
