Abstract:
Nonparametric forecasts based on local linearization of the shift map using a training data set were originally developed for low-dimensional chaotic systems.  In this research, jointly developed with Dimitris Giannakis and John Harlim, we make a rigorous connection between the shift map and the forward operator for ergodic stochastic differential equations on manifolds.  Estimating the forward operator gives us the ability to forecast full probability densities, and by representing these densities in a basis of smooth functions the forward operator becomes a matrix.  We show that the diffusion maps algorithm approximates the optimal basis for representing the forward operator.  However, the curse-of-dimensionality continues to restrict this nonparametric model to low-dimensional systems.  To address this issue we introduce two methods of lifting the nonparametric approach to high-dimensional problems.  In the first case, we assume that we have no knowledge of the model, but that we know the spatial structure of the data, allowing us to decompose the dynamics into Fourier modes for which nonparametric models can be built.  In the second case, we assume that a partial model is available, and that the unknown `model error' is low-dimensional in an appropriate sense.  We show that it is possible to extract and learn the unknown component of the model resulting in significantly improved forecasting.