This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks (NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nystrom-based interpolation method.

翻译：本文提出一个基于扩散图（DM）与深度学习的无网格计算框架及机器学习理论，用于在由点云识别的未知流形上求解椭圆型偏微分方程。该偏微分方程求解器被形式化为一个监督学习任务，通过求解最小二乘回归问题来建立逼近偏微分方程（及边界条件）的代数方程。该代数方程涉及通过DM渐近展开获得的类图拉普拉斯矩阵，该矩阵是二阶椭圆微分算子的相容估计量。所得数值方法可归结为求解一个高度非凸的经验风险最小化问题，其解来自神经网络（NN）假设空间。在适定椭圆偏微分方程设定下，当假设空间由具有无限宽度或深度的神经网络构成时，我们证明经验损失函数的全局极小值在大量训练数据极限下是相容解。当假设空间为两层神经网络时，我们证明在宽度充分大的条件下，梯度下降能够找到经验损失函数的全局极小值。支撑性数值算例展示了解的收敛性，涵盖从低余维/高余维简单流形到带有/不带边界的粗糙曲面。我们还证明，所提出的神经网络求解器能够在新数据点上对偏微分方程解进行鲁棒的泛化，其泛化误差几乎与训练误差相同，超越了基于Nyström插值的方法。