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. 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:本文提出一个无网格的计算框架和机器学习理论, 以解决在未知的方块上的椭圆 PDE, 以分布图( DM) 和深层学习为基础, 用点云确定。 PDE 求解器是设计成一个受监督的学习任务, 以解决最小方形回归问题, 使代数方程式( 如果适用的话, 以及边界条件 ) 。 此代数方程式包含一个通过 DM 自动扩展获得的图形- Laplaceacian 类型矩阵, 这是第二阶级椭圆差操作器的一致估计值。 由此产生的数字方法是解决高度非colvex 实证风险最小化问题, 由神经网络的假设空间解决一个最小化的平方形回归问题。 当假设空间由宽度或深度的神经网络组成时, 我们显示全球经验损失最小化功能是大型培训数据的限制。 当假设空间是双层固度固化的神经网络时, 我们显示一个足够大的宽度、 梯度跨度的实验风险最小化、 和高度的地层下层最小化的模型显示一个全球数据最小化、 显示一个来自普通的数值最小化的内流流流流解的数值最小化的模型。