  We present a deep learning model, DE-LSTM, for the simulation of a stochastic
process with an underlying nonlinear dynamics. The deep learning model aims to
approximate the probability density function of a stochastic process via
numerical discretization and the underlying nonlinear dynamics is modeled by
the Long Short-Term Memory (LSTM) network. It is shown that, when the numerical
discretization is used, the function estimation problem can be solved by a
multi-label classification problem. A penalized maximum log likelihood method
is proposed to impose a smoothness condition in the prediction of the
probability distribution. We show that the time evolution of the probability
distribution can be computed by a high-dimensional integration of the
transition probability of the LSTM internal states. A Monte Carlo algorithm to
approximate the high-dimensional integration is outlined. The behavior of
DE-LSTM is thoroughly investigated by using the Ornstein-Uhlenbeck process and
noisy observations of nonlinear dynamical systems; Mackey-Glass time series and
forced Van der Pol oscillator. It is shown that DE-LSTM makes a good prediction
of the probability distribution without assuming any distributional properties
of the stochastic process. For a multiple-step forecast of the Mackey-Glass
time series, the prediction uncertainty, denoted by the 95\% confidence
interval, first grows, then dynamically adjusts following the evolution of the
system, while in the simulation of the forced Van der Pol oscillator, the
prediction uncertainty does not grow in time even for a 3,000-step forecast.
