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：本文提出了一种基于扩散图（DM）和深度学习网格无关的计算框架与机器学习理论，用于求解以点云形式识别的未知流形上的椭圆型偏微分方程。该偏微分方程求解器被构建为监督学习任务，通过求解一个最小二乘回归问题来施加逼近偏微分方程（以及适用时的边界条件）的代数方程。该代数方程涉及一个通过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一致估计量。由此产生的数值方法需要求解一个高度非凸的经验风险最小化问题，该问题的解来自神经网络假设空间。在适定的椭圆型PDE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化子在大训练数据极限下是一个一致解。当假设空间为两层神经网络时，我们证明对于足够大的宽度，梯度下降能够找到经验损失函数的全局极小化子。支持性的数值示例展示了解的收敛性，涵盖从低维和高维共维的简单流形到带/不带边界的粗糙表面。我们还表明，所提出的神经网络求解器能够在新数据点上稳健地泛化PDE解，且泛化误差与训练误差几乎相同，优于基于Nyström的插值方法。