Kinetically Consistent Coarse Graining using Kernel-based Extended Dynamic Mode Decomposition

In this paper, we show how kernel-based models for the Koopman generator -- the gEDMD method -- can be used to identify coarse-grained dynamics on reduced variables, which retain the slowest transition timescales of the original dynamics. The centerpiece of this study is a learning method to identify an effective diffusion in coarse-grained space, which is similar in spirit to the force matching method. By leveraging the gEDMD model for the Koopman generator, the kinetic accuracy of the CG model can be evaluated. By combining this method with a suitable learning method for the effective free energy, such as force matching, a complete model for the effective dynamics can be inferred. Using a two-dimensional model system and molecular dynamics simulation data of alanine dipeptide and the Chignolin mini-protein, we demonstrate that the proposed method successfully and robustly recovers the essential kinetic and also thermodynamic properties of the full model. The parameters of the method can be determined using standard model validation techniques.