1. Data-driven stochastic model reduction for complex dynamics
Space-time reduction: dimension reduction in space and time reduction with large time-steppingMany problems in science and engineering involve nonlinear dynamical systems that are too complex or computationally expensive for full solutions, but one is interested mainly in a subset of the variables. Such problems appear, for example, in weather and climate modeling, in statistical mechanics, and in the mechanics of turbulent flow. In this setting, it is desirable to construct effective reduced models for the variables of interest using data that come either from observations or from fine numerical simulations. These reduced models are supposed to capture the key statistical and dynamical properties of the original systems, and therefore, stochastic reduced models are often preferred. Statistical inference and machine learning methods are natural tools for the construction of such reduced models for data. We focus on the mathematical understanding of such inference-based data-driven approach for model reduction of complex dynamics.
Discrete-time stochastic parametrization. In collaboration with Alexandre J. Chorin, I proposed a discrete-time parametrization framework to infer from discrete-time partial data a reduced model in the form of NARMAX (nonlinear autoregression moving average with exogenous input). This provides flexibility in the parametrization of memory effects as suggested by Mori-Zwanzig formalism, simplifies the inference from data and accounts for the discretization errors.
Parametrization of approximate inertial manifolds. The major challenge in NARMAX (and general semi-parametric) inference is to derive a model structure. Together with Kevin K. Lin, we developed a method for deriving structures by parametrizing approximate inertial manifolds. This method applies to dissipative systems with inertial manifolds such as the Kuramoto-Sivashinsky equation.
Model reduction by statistic learning We view the model reduction as a problem of learning the forward map of the (stochastic) process of interest, which is often the large-scale variables. The reduced model approximates the forward map optimal in a suitable hypothesis function space. Thus, we do not require the existence of an inertial manifold. The major issues are: 1. to learn the forward map, which is often high-dimensional; 2. to quantify the approximation of the stochastic process in distribution. To address 1, we focus on finding structures to reduce the complexity of function so that it can be learned via parametric inference or nonparametric inference of low-dimensional functions. Our efforts aim for a better understanding of the modeling and provide guidance on the development of machine learning techniques.
2. ISALT: inference-based schemes adaptive to large time-stepping
Reduction in time: focus the computational efforts on the large time scaleEfficient simulation of SDEs is essential in predictive modeling in multi-time scales. However, due to stiffness and accuracy requirements, most simulations have to forward with small time steps, thus being computationally costly. This is particularly true when the drift is non-global Lipschitz, and an implicit scheme is necessary to be stable and accurate. We introduce a discrete-time flow map approximation framework for inferring schemes adaptive to large time-stepping.