Deep learning algorithm for data-driven simulation of noisy dynamical system
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We present a deep learning model, DE-LSTM, for the simulation of a stochastic process with 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. After the numerical discretization by a softmax function, the function estimation problem is solved by a multi-label classification problem. A penalized maximum log likelihood method is proposed to impose smoothness in the predicted probability distribution. It is shown that LSTM is a state space model, where the internal dynamics consists of a system of relaxation processes. A sequential Monte Carlo method is outlined to compute the time evolution of the probability distribution. The behavior of DE-LSTM is investigated by using the Ornstein-Uhlenbeck process and noisy observations of Mackey-Glass equation and forced Van der Pol oscillators. While the probability distribution computed by the conventional maximum log likelihood method makes a good prediction of the first and second moments, the Kullback-Leibler divergence shows that the penalized maximum log likelihood method results in a probability distribution closer to the ground truth. It is shown that DE-LSTM makes a good prediction of the probability distribution without assuming any distributional properties of the noise. For a multiple-step forecast, it is found that the prediction uncertainty, denoted by the 95 system, Mackey-Glass time series, the 95 exhibits an oscillatory behavior, instead of growing indefinitely, while for the forced Van der Pol oscillator, the prediction uncertainty does not grow in time even for 3,000-step forecast.
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