  We address the problem of constructing accurate mathematical models of the
dynamics of complex systems projected on a collective variable. To this aim we
introduce a conceptually simple yet effective algorithm for estimating the
parameters of Langevin and Fokker-Planck equations from a set of short,
possibly out-of-equilibrium molecular dynamics trajectories, obtained for
instance from transition path sampling or as relaxation from high free-energy
configurations. The approach maximizes the model likelihood based on any
explicit expression of the short-time propagator, hence it can be applied to
different evolution equations. We demonstrate the numerical efficiency and
robustness of the algorithm on model systems, and we apply it to reconstruct
the projected dynamics of pairs of C60 and C240 fullerene molecules in explicit
water. Our methodology allows reconstructing the accurate thermodynamics and
kinetics of activated processes, namely free energy landscapes, diffusion
coefficients, and kinetic rates. Compared to existing enhanced sampling
methods, we directly exploit short unbiased trajectories at a competitive
computational cost.
