Learning the reaction coordinate: collective variables from physical intuition to generative models.

Collective variables (CVs) are low-dimensional projections of molecular configuration space that serve a dual purpose: they provide mechanistic interpretability by distilling complex transformations into comprehensible reaction coordinates, and they underpin the majority of enhanced sampling methods by defining the directions along which free-energy barriers are overcome. The discovery of suitable CVs has evolved from reliance on chemical intuition, for instance by selecting distances, angles, or dihedrals by hand, the systematic linear approaches, such as principal component analysis and time-lagged independent component analysis, to nonlinear manifold-learning techniques including diffusion maps and variational autoencoders. Deep-learning methods have further expanded this landscape: committor-based neural networks approximate the optimal reaction coordinate from trajectory data, discriminant and variational models learn CVs tied to metastable-state separation or slow kinetics, and equivariant graph neural networks construct symmetry-preserving representations directly from atomic coordinates. In parallel, generative models-normalizing flows, diffusion-based samplers, and learned transfer operators-have begun to bypass explicit dimensionality reduction altogether, learning equilibrium distributions or dynamical propagators in the full configurational space. Yet, these models encode latent structure from which CVs can be extracted