LDE-Net: Lévy Induced Stochastic Differential Equation Equipped with Neural Network for Time Series Forecasting

With the fast development of modern deep learning techniques, the study of dynamic systems and neural networks is increasingly benefiting each other in a lot of different ways. Since uncertainties often arise in real world observations, SDEs (stochastic differential equations) come to play an important role in scientific modeling. To this end, we employ a collection of SDEs with drift and diffusion terms approximated by neural networks to predict the trend of chaotic time series which has big jump properties. Our contributions are, first, we propose LDE-Net, which explores compounded SDEs with $\alpha$-stable L\'evy motion to model complex time series data and solve the problem through neural network approximation. Second, we theoretically prove the convergence of our algorithm with respect to hyper-parameters of the neural network, and obtain the error bound without curse of dimensionality. Finally, we illustrate our method by applying it to real time series data and find the accuracy increases through the use of non-Gaussian L\'evy processes. We also present detailed comparisons in terms of data patterns, various models, different shapes of L\'evy motion and the prediction lengths.