分析深度柯尔莫哥洛夫方法解热方程的误差收敛性
Error analysis for the deep Kolmogorov method
- 基于随机优化，分析神经网络结构大小对误差影响
- 给出网络深度、宽度及采样点数量与误差的定量关系
- 适合研究深度学习解PDE理论的学者参考
深度柯尔莫哥洛夫方法是一种简单且流行的基于深度学习的柯尔莫哥洛夫型偏微分方程（PDE）近似求解方法。本文针对热方程提供了该方法的误差分析。具体而言，揭示了精确解与近似神经网络实现函数之间整体均方误差的收敛性及其收敛速率，该结果以近似网络的架构规模（深度/隐藏层层数、宽度/隐藏层节点数）、损失函数中使用的随机采样点数量（即损失函数中的输入-输出数据对数量），以及所用随机优化方法造成的优化误差为变量。分析表明，随着网络规模增大、采样点增多和优化误差减小，误差呈可量化下降趋势。
原文摘要 · Abstract (English)
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.
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