Differentiable, model-agnostic free energy calculation

The vibrational free energy is essential to predict finite temperature material properties. Current methods employ slow, largely sequential sampling with a fixed machine learning interatomic potential (MLIP) to satisfy the tight 1-2meV/atom (1/40-1/20 kcal/mol) convergence requirements. Forward or back propagation of MLIP parameters is not practically possible, meaning estimates cannot be used in objective functions for alignment to reference data or distillation. For the broad class of generalized linear MLIPs we show free energies can be cast as the Legendre transform of a high-dimensional descriptor entropy, accurately estimated via score matching. Our main result is a model-agnostic estimator which returns meV/atom accurate, end-to-end free energies as a function of MLIP parameters. Sampling is efficient and highly parallel, requiring 10x fewer force calls and 100-1000x less walltime than a single thermodynamic integration estimate. Tensor compression allows lightweight storage and inference is instantaneous. In forward propagation, a single estimator predicts a broad ensemble high temperature thermodynamic integration calculations for W. In back-propagation, we fine-tune the $\alpha-\gamma$ transition temperature in an Fe model from 2000K to 1063K, a first demonstration of MLIP alignment against known phase boundaries.