  We present a deep neural network for a model-free prediction of a chaotic
dynamical system from noisy observations. The proposed deep learning model aims
to predict the conditional probability distribution of a state variable. The
Long Short-Term Memory network (LSTM) is employed to model the nonlinear
dynamics and a softmax layer is used to approximate a probability distribution.
The LSTM model is trained by minimizing a regularized cross-entropy function.
The LSTM model is validated against delay-time chaotic dynamical systems,
Mackey-Glass and Ikeda equations. It is shown that the present LSTM makes a
good prediction of the nonlinear dynamics by effectively filtering out the
noise. It is found that the prediction uncertainty of a multiple-step forecast
of the LSTM model is not a monotonic function of time; the predicted standard
deviation may increase or decrease dynamically in time.
