Galerkin Approximation of Dynamical Quantities using Trajectory Data

• Published in: arXiv.org 2019-02
• Main Authors: Thiede, Erik H, Giannakis, Dimitrios, Dinner, Aaron R, Weare, Jonathan
• Format: Article
• Language: English
• Subjects: Boundary conditions; Boundary value problems; Embedded systems; Galerkin method; Lag time; Markov chains; Model accuracy; Organic chemistry; Response time
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1810.01841

Understanding chemical mechanisms requires estimating dynamical statistics such as expected hitting times, reaction rates, and committors. Here, we present a general framework for calculating these dynamical quantities by approximating boundary value problems using dynamical operators with a Galerkin expansion. A specific choice of basis set in the expansion corresponds to estimation of dynamical quantities using a Markov state model. More generally, the boundary conditions impose restrictions on the choice of basis sets. We demonstrate how an alternative basis can be constructed using ideas from diffusion maps. In our numerical experiments, this basis gives results of comparable or better accuracy to Markov state models. Additionally, we show that delay embedding can reduce the information lost when projecting the system's dynamics for model construction; this improves estimates of dynamical statistics considerably over the standard practice of increasing the lag time.