Gradient Convergence of Deep Learning-Based Numerical Methods for BSDEs

• 来源期刊: 数学年刊B辑（英文版）
• 年，卷（期）: 2021,（2）
• 页码范围: 199-216
• 页数: 18页
• 语种: 英文
• DOI: 10.1007/s11401-021-0253-x

摘要

The authors prove the gradient convergence of the deep learning-based nu-merical method for high dimensional parabolic partial differential equations and backward stochastic differential equations,which is based on time discretization of stochastic differ-ential equations(SDEs for short)and the stochastic approximation method for noncon-vex stochastic programming problem.They take the stochastic gradient decent method,quadratic loss function,and sigmoid activation function in the setting of the neural net-work.Combining classical techniques of randomized stochastic gradients,Euler scheme for SDEs,and convergence of neural networks,they obtain the O(K-1/4)rate of gradient convergence with K being the total number of iterative steps.