Galerkin approximation of dynamical quantities using trajectory data.

Area of Science:

• Chemical dynamics
• Computational chemistry
• Statistical mechanics

Background:

• Estimating dynamical statistics (e.g., reaction rates, committors) is crucial for understanding chemical mechanisms.
• Current methods often rely on approximations and can be limited by model construction techniques.

Purpose of the Study:

• To present a general framework for calculating dynamical statistics by approximating boundary value problems.
• To introduce an alternative basis construction using diffusion maps for improved accuracy.
• To demonstrate the benefits of delay embedding in dynamical system projection for model construction.

Main Methods:

• Approximation of boundary value problems using dynamical operators with Galerkin expansion.
• Construction of an alternative basis set inspired by diffusion maps.
• Application of delay embedding for system dynamics projection.

Main Results:

• The proposed framework provides accurate estimations of dynamical statistics.
• The diffusion map-based basis set yields results comparable or superior to traditional Markov state models.
• Delay embedding significantly reduces information loss, enhancing dynamical statistics estimates.

Conclusions:

• The developed framework offers a robust and accurate approach to chemical mechanism analysis.
• The combination of Galerkin expansion, diffusion maps, and delay embedding presents a powerful computational tool.
• This work advances the accurate estimation of key dynamical quantities in chemical systems.