Xu Yong, Zhang Hao, Li Yongge, Zhou Kuang, Liu Qi, Kurths Jürgen
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China.
Center for Mathematical Sciences and School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.
Chaos. 2020 Jan;30(1):013133. doi: 10.1063/1.5132840.
The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed algorithm does not require any interpolation and coordinate transformation, which is different from the traditional numerical methods. The main novelty of this paper is that penalty factors are introduced to overcome the local optimization for the deep learning approach, and the corresponding setting rules are given. Meanwhile, we consider a normalization condition as a supervision condition to effectively avoid that the trial solution is zero. Several numerical examples are presented to illustrate performances of the proposed algorithm, including one-, two-, and three-dimensional systems. All the results suggest that the deep learning is quite feasible and effective to calculate the FP equation. Furthermore, influences of the number of hidden layers, the penalty factors, and the optimization algorithm are discussed in detail. These results indicate that the performances of the machine learning technique can be improved through constructing the neural networks appropriately.
随机微分方程的概率密度函数由福克 - 普朗克（FP）方程控制。基于深度神经网络开发了一种新颖的机器学习方法来求解一般的FP方程。所提出的算法不需要任何插值和坐标变换，这与传统数值方法不同。本文的主要新颖之处在于引入惩罚因子以克服深度学习方法的局部优化问题，并给出了相应的设置规则。同时，我们将归一化条件作为监督条件，以有效避免试验解为零的情况。给出了几个数值例子来说明所提算法的性能，包括一维、二维和三维系统。所有结果表明，深度学习在计算FP方程方面是相当可行且有效的。此外，还详细讨论了隐藏层数、惩罚因子和优化算法的影响。这些结果表明，通过适当地构建神经网络可以提高机器学习技术的性能。
