In this paper, we extend the Generalized Finite Difference Method (GFDM) on unknown compact submanifolds of the Euclidean domain, identified by randomly sampled data that (almost surely) lie on the interior of the manifolds. Theoretically, we formalize GFDM by exploiting a representation of smooth functions on the manifolds with Taylor's expansions of polynomials defined on the tangent bundles. We illustrate the approach by approximating the Laplace-Beltrami operator, where a stable approximation is achieved by a combination of Generalized Moving Least-Squares algorithm and novel linear programming that relaxes the diagonal-dominant constraint for the estimator to allow for a feasible solution even when higher-order polynomials are employed. We establish the theoretical convergence of GFDM in solving Poisson PDEs and numerically demonstrate the accuracy on simple smooth manifolds of low and moderate high co-dimensions as well as unknown 2D surfaces. For the Dirichlet Poisson problem where no data points on the boundaries are available, we employ GFDM with the volume-constraint approach that imposes the boundary conditions on data points close to the boundary. When the location of the boundary is unknown, we introduce a novel technique to detect points close to the boundary without needing to estimate the distance of the sampled data points to the boundary. We demonstrate the effectiveness of the volume-constraint employed by imposing the boundary conditions on the data points detected by this new technique compared to imposing the boundary conditions on all points within a certain distance from the boundary, where the latter is sensitive to the choice of truncation distance and require the knowledge of the boundary location.

翻译：本文在欧几里得空间中的未知紧致子流形上推广了广义有限差分法(GFDM)，其中流形由随机采样（几乎必然）位于其内部的数据点进行识别。理论上，我们通过利用切丛上多项式泰勒展开对流形上的光滑函数进行表示，从而形式化GFDM。我们通过逼近拉普拉斯-贝尔特拉米算子来阐释该方法，其中通过结合广义移动最小二乘算法与新颖的线性规划方法实现稳定逼近：该线性规划松弛了对角占优约束条件，使得即使采用高阶多项式也能获得可行解。我们建立了GFDM在求解泊松偏微分方程时的理论收敛性，并通过数值实验验证了该方法在低维、中等高余维以及未知二维曲面上的简单光滑流形上的精度。针对无边界数据点的狄利克雷泊松问题，我们采用体积约束方法将边界条件施加于靠近边界的数据点上。当边界位置未知时，我们提出了一种新技术，无需估计采样数据点到边界的距离即可检测靠近边界的点。相较于在距离边界一定范围内的所有点上施加边界条件的方法（该方法对截断距离的选择敏感且需要已知边界位置），我们展示了通过新检测技术施加边界条件的体积约束方法的有效性。