  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 numercial 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- and two-dimensional systems. All the results
suggest that the deep learning is quite feasible and effective to calculate the
FP equation. Further, 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.
