Galerkin Approximation of Dynamical Quantities using Trajectory Data

TL;DR

This paper introduces a flexible Galerkin-based framework for estimating dynamical statistics from trajectory data, improving accuracy over traditional Markov state models through alternative basis construction and delay embedding techniques.

Contribution

It develops a general Galerkin approximation method for dynamical quantities, incorporating diffusion maps and delay embedding to enhance estimation accuracy.

Findings

Alternative basis sets can outperform Markov state models in accuracy.

Delay embedding significantly improves dynamical statistics estimation.

The proposed framework is versatile and adaptable to different basis choices.

Abstract

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.