A data-driven framework for sparsity-enhanced surrogates with arbitrary mutually dependent randomness

Petrov-Galerkin methods for the construction of non-Markovian dynamics preserving nonlocal statistics

A common observation in coarse-graining a molecular system is the non-Markovian behavior, primarily due to the lack of scale separations. This is reflected in the strong memory effect and the non-white noise spectrum, which must be incorporated into a coarse-grained description to correctly predict dynamic properties. To construct a stochastic model that gives rise to the correct non-Markovian dynamics, we propose a Galerkin projection approach, which transforms the exhausting effort of finding an appropriate model to choosing appropriate subspaces in terms of the derivatives of the coarse-grained variables and, at the same time, provides an accurate approximation to the generalized Langevin equation. We introduce the notion of fractional statistics that embodies nonlocal properties. More importantly, we show how to pick subspaces in the Galerkin projection so that those statistics are automatically matched.

Data-driven molecular modeling with the generalized Langevin equation

The complexity of molecular dynamics simulations necessitates dimension reduction and coarse-graining techniques to enable tractable computation. The generalized Langevin equation (GLE) describes coarse-grained dynamics in reduced dimensions. In spite of playing a crucial role in non-equilibrium dynamics, the memory kernel of the GLE is often ignored because it is difficult to characterize and expensive to solve. To address these issues, we construct a data-driven rational approximation to the GLE. Building upon previous work leveraging the GLE to simulate simple systems, we extend these results to more complex molecules, whose many degrees of freedom and complicated dynamics require approximation methods. We demonstrate the effectiveness of our approximation by testing it against exact methods and comparing observables such as autocorrelation and transition rates.