标题： 深度柯尔莫哥洛夫方法的误差分析 显示英文标题

标题： Error analysis for the deep Kolmogorov method

摘要： 深度科尔莫戈罗夫方法是一种简单且流行的基于深度学习的方法，用于近似求解科尔莫戈罗夫类型的偏微分方程（PDE）。在本工作中，我们提供了针对热方程PDE的深度科尔莫戈罗夫方法的误差分析。具体来说，我们揭示了精确解与与随机优化算法相关的近似深度神经网络（DNN）实现函数之间的整体均方距离的收敛性及其收敛速率，这些速率是根据近似DNN的结构大小（深度/隐藏层的数量和隐藏层的宽度）、损失函数中使用的随机样本点数量（损失函数中使用的输入输出数据对的数量）以及所采用的随机优化方法产生的优化误差大小来衡量的。 显示英文摘要

摘要： The deep Kolmogorov method is a simple and popular deep learning based method for approximating solutions of partial differential equations (PDEs) of the Kolmogorov type. In this work we provide an error analysis for the deep Kolmogorov method for heat PDEs. Specifically, we reveal convergence with convergence rates for the overall mean square distance between the exact solution of the heat PDE and the realization function of the approximating deep neural network (DNN) associated with a stochastic optimization algorithm in terms of the size of the architecture (the depth/number of hidden layers and the width of the hidden layers) of the approximating DNN, in terms of the number of random sample points used in the loss function (the number of input-output data pairs used in the loss function), and in terms of the size of the optimization error made by the employed stochastic optimization method.