Learning collective variables that respect permutational symmetry

In addition to translational and rotational symmetries, clusters of identical interacting particles possess permutational symmetry. Coarse-grained models for such systems are instrumental in identifying metastable states, providing an effective description of their dynamics, and estimating transition rates. We propose a numerical framework for learning collective variables that respect translational, rotational, and permutational symmetries, and for estimating transition rates and residence times. It combines a sort-based featurization, residence manifold learning in the feature space, and learning collective variables with autoencoders whose loss function utilizes the orthogonality relationship (Legoll and Lelievre, 2010). The committor of the resulting reduced model is used as the reaction coordinate in the forward flux sampling and to design a control for sampling the transition path process. We offer two case studies, the Lennard-Jones-7 in 2D and the Lennard-Jones-8 in 3D. The transition rates and residence times computed with the aid of the reduced models agree with those obtained via brute-force methods.