Model-free prediction of noisy chaotic time series by deep learning
This work addresses the challenge of forecasting chaotic systems from noisy data for applications in fields like physics or engineering, but it is incremental as it applies an existing LSTM method to this specific problem.
The authors tackled the problem of predicting noisy chaotic time series without a model by using a deep neural network, specifically an LSTM with a softmax layer, and achieved good prediction performance by effectively filtering out noise in systems like Mackey-Glass and Ikeda equations, with the prediction uncertainty showing non-monotonic dynamic changes over time.
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.