非线性Klein-Gordon 方程是一类重要的偏微分方程，可以利用嵌套皮卡迭代积分器(NPI) 求解其数值解格式，本文主要研究如何利用机器学习求得高阶NPI 数值解格式的近似结果。本文首先讲解了长短期记忆(LSTM) 神经网络的基本原理，并以此为核心构建神经网络学习器，之后介绍了迁移学习和AdaBoost.R2 算法，详细阐述了结合了二者的Two-stage TrAdaBoost.R2 算法，最后分别调用神经网络学习器和Two-stage TrAdaBoost.R2 算法对3 阶NPI 数值解格式进行数值实验。实验结果表明，Two-stage TrAdaBoost.R2 算法在采用大量多源低阶（1 阶、2 阶）数据，少量3 阶数据时有更优异的表现，且拟合效果随着3 阶数据量的增大而提升。但是当采用大量3 阶数据时，神经网络学习器具有更好的表现。

The nonlinear Klein-Gordon equation is an important class of partial differential equations, which can be numerically solved by the nested Picard iterative integrator( NPI ). The object of this paper is to study how to use machine learning to obtain the approximate results of high-order NPI numerical solution. In this paper, the basic principle of Long Short-Term Memory(LSTM) neural network is explained and LSTM is used to construct a neural network learner as the core. Then this paper introduces the transfer learning and AdaBoost.R2 algorithm and elaborates the Two-stage
TrAdaBoost.R2 algorithm which combines the two. Finally, the neural network learner and the Two-stage TrAdaBoost.R2 algorithm are used to perform numerical experiments on the third-order NPI numerical solution. The experimental results show that the Two-stage TrAdaBoost.R2 algorithm has better performance when using a large number of multi-source low-order (first-order, second order) data and a small number of third-order data, and the fitting effect increases with the increase of high-order data. However, when a large amount of third-order data is used, the
neural network learner has better performance.