Data-driven Modeling of Equilibrium and Non-equilibrium Statistics

An important issue in applied and computational sciences is to find the essential reduced models to predict variables of interests from high-dimensional complex dynamical systems. Given our advanced capability to collect big data, an important challenge is to leverage the information carried by the data to improve the modeling effort. Computationally, this requires adequate inference of appropriate parameters such that their uncertainties are quantifiable. A much more challenging yet important issue is to be able to make prediction in the presence of external disturbances. This problem has a wide range of applications such as in climate change science where one is interested to predict the climate change statistics corresponding to exogenous forcing such as the volcanic eruptions or even the anthropogenic factor such as the human activities. The projects in this proposal are to address these issues. While the developed methodology is aimed for general modeling of multi-scale phenomena, our focus will be to improve the understanding and prediction of the deformation behavior of graphene.

Two projects are proposed: 1. Data-driven reduced modeling paradigms to capture coarse grained statistical solutions of the underlying dynamics. The methodology involves the Mori-Zwanzig formalism, a precise description of the memory effect to take into account the interactions between processes occurring on different physical scales, and a data-driven numerical scheme for estimating the parameters of the stochastic reduced model. 2. Estimation of parameters in the reduced models to predict changes on the statistical solutions in the presence of small external disturbances. This project involves employing the Padè approximation on appropriate integral operators and designing efficient algorithm to solve a system of nonlinear equations that respect appropriate equilibrium statistics of the unperturbed data, leveraging the formulation from the fluctuation-dissipation theory.