In this paper, we extend the Generalized Moving Least-Squares (GMLS) method in two different ways to solve the vector-valued PDEs on unknown smooth 2D manifolds without boundaries embedded in $\mathbb{R}^{3}$, identified with randomly sampled point cloud data. The two approaches are referred to as the intrinsic method and the extrinsic method. For the intrinsic method which relies on local approximations of metric tensors, we simplify the formula of Laplacians and covariant derivatives acting on vector fields at the base point by calculating them in a local Monge coordinate system. On the other hand, the extrinsic method formulates tangential derivatives on a submanifold as the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. One challenge of this method is that the discretization of vector Laplacians yields a matrix whose size relies on the ambient dimension. To overcome this issue, we reduce the dimension of vector Laplacian matrices by employing an appropriate projection. The complexity of both methods scales well with the dimension of manifolds rather than the ambient dimension. We also present supporting numerical examples, including eigenvalue problems, linear Poisson equations, and nonlinear Burgers' equations, to examine the numerical accuracy of proposed methods on various smooth manifolds.

翻译：本文从两个不同方向扩展了广义移动最小二乘法（GMLS），用于求解嵌入在 $\mathbb{R}^{3}$ 中、由随机采样点云数据识别的无边界光滑二维未知流形上的向量值偏微分方程。这两种方法分别称为内蕴法和外蕴法。对于依赖度量张量局部逼近的内蕴法，我们通过在局部 Monge 坐标系中计算，简化了作用于基点处向量场的拉普拉斯算子与协变导数公式。另一方面，外蕴法将子流形上的切向导数表述为环境欧氏空间中方向导数向子流形切空间的投影。该方法的一个挑战在于向量拉普拉斯算子的离散化会产生一个矩阵，其维度依赖于环境空间的维数。为解决此问题，我们通过采用适当的投影降低了向量拉普拉斯矩阵的维度。两种方法的计算复杂度均与流形本身的维度（而非环境空间维度）保持良好的比例关系。我们还提供了包括特征值问题、线性泊松方程和非线性 Burgers 方程在内的数值算例，以检验所提方法在不同光滑流形上的数值精度。