Committor-regularized learning of differentiable collective variables from non-differentiable structural descriptors

Abstract

Collective variables (CVs) are essential for interpreting and accelerating rare events in molecular simulations, yet their construction is constrained by the requirement of differentiability with respect to atomic coordinates. This excludes many powerful structural descriptors, such as discrete local-structure classifiers, which are routinely used for analysis but cannot be applied directly to define bias potentials in enhanced-sampling molecular dynamics simulations. Here, we introduce a physics-informed machine-learning framework that transforms non-differentiable structural descriptors into fully differentiable, bias-ready collective variables. The approach uses graph neural networks to learn smooth surrogate representations of inherently discontinuous descriptors, trained directly from atomic coordinates while preserving their physical interpretability. To guide interpolation between metastable states, we introduce committor regularization, a physically motivated constraint derived from transition path theory that biases learning toward committor-like behavior along reactive pathways. We apply this framework to crystal nucleation in a supercooled copper melt, using polyhedral template matching (PTM), a discrete, non-differentiable classifier of local crystal structure, as the target descriptor. Neural network CVs are trained to reproduce the fractions of FCC and BCC environments inferred from PTM labels while remaining fully differentiable. We show that committor-regularized models yield reproducible free-energy surfaces across ensembles of independently trained CVs, overcoming the stochastic variability that commonly limits machine-learned collective variables. By bridging discrete physics-based descriptors and continuous differentiable representations, this work establishes a general route to constructing interpretable, reproducible, and physically grounded collective variables for enhanced sampling.