DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:**Tyrus Berry, Pennsylvania State University/George Mason
University

**Title: ***
Data-driven forecasting without a model and with a partially known model
*

**Abstract:**
The idea of data-driven forecasting is to use a training data set to make predictions without specifying a model for the dynamics. This approach is analogous to the histogram in statistics, where the shape of the probability distribution is determined directly from the data. Data-driven forecasting based on local linear interpolation was originally developed for low-dimensional chaotic systems. In this research, jointly developed with Dimitris Giannakis and John Harlim, we introduce a data-driven method which builds a global forecasting model for a large class of stochastic dynamical systems. For these systems, the evolution of a probability distribution is given by a linear operator (specifically, the semigroup solution of the Fokker-Planck PDE). Using a basis of smooth functions, we represent the linear forecasting operator as a matrix. By simply applying this matrix repeatedly, we produce videos showing the evolution of an initial distribution, which we compare to an ensemble which evolves according to the true dynamical system. However, for practical applications the data-driven approach is restricted to low-dimensional systems due to the curse-of-dimensionality. To address this issue, 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 improved forecasts.

**Time:** Friday, February 27, 2015, 1:30-2:30 p.m.

**Place:** Exploratory Hall, Room 4106

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491