Title: Stochastic model reduction of nonlinear dynamics by parametric inference
Abstract: The construction of effective statistical-dynamical models for high-dimensional nonlinear dynamics is one of the major challenges in computational science and engineering. With the exponentially increasing observational or simulated data, statistical learning approaches provide promising tools to address the challenge. A fundamental idea in these approaches is to approximate the discrete-time flow map of the dynamics. A major difficulty is the curse of dimensionality (COD) because the flow map is often high or infinite-dimensional. To overcome the COD, we investigate a semi-parametric approach that derives parametric models from the field knowledge such as mathematical theory and numerical schemes. With efficient parametric forms, we estimate the parameters by least squares and show the convergence of the estimators for ergodic systems. We show that this approach leads to effective statistical-dynamical models for deterministic and stochastic (partial) differential equations, such as the Kuramoto-Sivashisky equation to stochastic Burgers equations. In particular, we highlight the shift from the nonlinear Galerkin method to statistical inference, and discuss an open question on optimal space-time reduction.