Prediction of high-dimensional nonlinear dynamic systems is often difficult when only partial observations are available, because such systems are often expensive to solve in full and the initial data will be incomplete. The development of reduced models for the observed variables is thus needed. The challenges come from the nonlinear interactions between the observed variables and the unobserved variables, and the difficulties in quantifying uncertainties from discrete data.
We address these challenges by developing discrete nonlinear stochastic reduced systems, in which one formulates discrete solvable approximate equations for the observed variables, and uses data and statistical methods to account for the impact of the unobserved variables. A key ingredient in the construction of the stochastic reduced systems is a discrete-time stochastic parametrization based on inference of nonlinear time series. We demonstrate our approach with some model problems, including the Lorenz 96 system and the Kuramoto-Sivashinsky equation. This is joint work with Alexandre Chorin and Kevin Lin.