Variational approach for learning Markov processes from time series data


主講人:吳昊 同濟大學數學科學學院教授




內容介紹:Inference, prediction and control of complex dynamical systems from time series  is important in many areas, including financial markets, power grid management,  climate and weather modeling, or molecular dynamics. The analysis of such highly  nonlinear dynamical systems is facilitated by the fact that we can often find a  (generally nonlinear) transformation of the system coordinates to features in  which the dynamics can be excellently approximated by a linear Markov model.  Moreover, the large number of system variables often change collectively on  large time- and length-scales, facilitating a low-dimensional analysis in  feature space. In this paper, we introduce a variational approach for Markov  processes (VAMP) that allows us to find optimal feature mappings and optimal  Markovian models of the dynamics from given time series data. The key insight is  that the best linear model can be obtained from the top singular components of  the Koopman operator. This leads to the definition of a family of score  functions called VAMP-r which can be calculated from data, and can be employed  to optimize a Markovian model. In addition, based on the relationship between  the variational scores and approximation errors of Koopman operators, we propose  a new VAMP-E score, which can be applied to cross-validation for hyper-parameter  optimization and model selection in VAMP. VAMP is valid for both reversible and  nonreversible processes and for stationary and non-stationary processes or  realizations.