The inverse problem in statistical physics asks: given an observed set of properties, what are the molecules, energy functions, and/or thermodynamic conditions that give rise to them? Such a task is at the heart of multiscale strategies, in which one desires a coarse-grained or low resolution model that faithfully reproduces the properties of a detailed one. There have been several specific approaches to this problem, including the reverse Monte Carlo method [Lyubartsev and Laaksonen, Phys. Rev. E 52, 3730 (1995)] and the force-matching approach [Ercolessi and Adams, Europhys. Lett. 26, 583 (1994); Voth and Izvekov, J. Phys. Chem. B 109, 2469 (2005)]. However, it has been challenging to identify general theoretical approaches, for arbitrary systems, energy functions, and ensembles.

Here we show that the relative entropy, Srel = sum p_T ln(p_T/ p_M), provides a fundamental overlap metric and unifying framework for inverse molecular-thermodynamic problems, involving optimization of a model system (`M') to reproduce the properties of a target one (`T'). We demonstrate that the relative entropy serves as a generating function for principles in variational mean field theory and uniqueness, and gives intuitive results for simple case scenarios in model development. Importantly, we show that the relative entropy provides new numerical techniques for linking models at different resolutions, by coupling its determination with flat-histogram and transition-matrix-based Monte Carlo algorithms. We also show that the relative entropy carries physical significance by using it to quantify the deviations of a three-site model of water from simple liquids; importantly, we demonstrate a surprising close connection between the relative entropy and kinetic anomalies in water. We then use the relative entropy to find optimal simple and conformal models that represent complex fluids and mixtures.