We discuss techniques for studying, in a quantitative fashion, certain properties of high-dimensional dynamical systems in view of performing model reduction, while preserving short and large time properties of the system. In particular, in the context of molecular dynamics we will discuss techniques for estimating, in a robust fashion, an effective number of degrees of freedom of the system, which may vary in the state space of the system, and a local scale where the dynamics is well-approximated by a reduced dynamics with a small number of degrees of freedom. We use these ideas in two ways: (1) given long trajectories of the system, to produce an approximation to the propagator of the system and obtain reaction coordinates for the system that capture the large time behavior of the dynamics; (2) to learn, given local short parallel simulations, a family of local approximations to the system, that can be pieced together to form a fast global reduced model for the system, for which we can guarantee (under suitable assumptions) that large time accuracy is bounded by the small time accuracy of the local simulators. We discuss applications to homogenization of rough diffusions in low and high dimensions.
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