# DE-RNN: Forecasting the Probability Density Function of Nonlinear Time Series

## Abstract

Model-free identification of a nonlinear dynamical system from the noisy observations is of current interest due to its direct relevance to many applications in Industry 4.0. Making a prediction of such noisy time series constitutes a problem of learning the nonlinear time evolution of a probability distribution. Capability of most of the conventional time series models is limited when the underlying dynamics is nonlinear, multi-scale or when there is no prior knowledge at all on the system dynamics. We propose DE-RNN (Density Estimation Recurrent Neural Network) to learn the probability density function (PDF) of a stochastic process with an underlying nonlinear dynamics and compute the time evolution of the PDF for a probabilistic forecast. A Recurrent Neural Network (RNN)-based model is employed to learn a nonlinear operator for the temporal evolution of the stochastic process. We use a softmax layer for a numerical discretization of a smooth PDF, which transforms a function approximation problem to a classification task. A regularized cross-entropy method is introduced to impose a smoothness condition on the estimated probability distribution. A Monte Carlo procedure to compute the temporal evolution of the distribution for a multiple-step forecast is presented. It is shown that the proposed algorithm can learn the nonlinear multi-scale dynamics from the noisy observations and provides an effective tool to forecast time evolution of the underlying probability distribution. Evaluation of the algorithm on three synthetic and two real data sets shows advantage over the compared baselines, and a potential value to a wide range of problems in physics and engineering.