With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics (SINDy) has been introduced to identify the governing equations of dynamical systems from simulation data. In this study, we extend SINDy to stochastic dynamical systems which are frequently used to model biophysical processes. We prove the asymptotic correctness of stochastic SINDy in the infinite data limit, both in the original and projected variables. We discuss algorithms to solve the sparse regression problem arising from the practical implementation of SINDy and show that cross validation is an essential tool to determine the right level of sparsity. We demonstrate the proposed methodology on two test systems, namely, the diffusion in a one-dimensional potential and the projected dynamics of a two-dimensional diffusion process.

Identification of the collective coordinates that describe rare events in complex molecular transitions such as protein folding has been a key challenge in the theoretical molecular sciences. In the Diffusion Map approach, one assumes that the molecular configurations sampled have been generated by a diffusion process, and one uses the eigenfunctions of the corresponding diffusion operator as reaction coordinates. While diffusion coordinates (DCs) appear to provide a good approximation to the true dynamical reaction coordinates, they are not parametrized using dynamical information. Thus, their approximation quality could not, as yet, be validated, nor could the diffusion map eigenvalues be used to compute relaxation rate constants of the system. Here we combine the Diffusion Map approach with the recently proposed Variational Approach for Conformation Dynamics (VAC). Diffusion Map coordinates are used as a basis set, and their optimal linear combination is sought using the VAC, which employs time-correlation information on the molecular dynamics (MD) trajectories. We have applied this approach to ultra-long MD simulations of the Fip35 WW domain and found that the first DCs are indeed a good approximation to the true reaction coordinates of the system, but they could be further improved using the VAC. Using the Diffusion Map basis, excellent approximations to the relaxation rates of the system are obtained. Finally, we evaluate the quality of different metric spaces and find that pairwise minimal root-mean-square deviation performs poorly, while operating in the recently introduced kinetic maps based on the time-lagged independent component analysis gives the best performance.

Coarse-graining has become an area of tremendous importance within many different research fields. For molecular simulation, coarse-graining bears the promise of finding simplified models such that long-time simulations of large-scale systems become computationally tractable. While significant progress has been made in tuning thermodynamic properties of reduced models, it remains a key challenge to ensure that relevant kinetic properties are retained by coarse-grained dynamical systems. In this study, we focus on data-driven methods to preserve the rare-event kinetics of the original system, and make use of their close connection to the low-lying spectrum of the system's generator. Building on work by Crommelin and Vanden-Eijnden, SIAM Multiscale Model. Simul. (2011), we present a general framework, called spectral matching, which directly targets the generator's leading eigenvalue equations when learning parameters for coarse-grained models. We discuss different parametric models for effective dynamics and derive the resulting data-based regression problems. We show that spectral matching can be used to learn effective potentials which retain the slow dynamics, but also to correct the dynamics induced by existing techniques, such as force matching.