Data-Driven Classical Density Functional Theory: A Case for Physics Informed Learning

In traditional sense, physical modeling is often associated with analytic derivations, followed by computation and validation against data. On the other hand, modern statistical inference offers principled means to accomplish similar goals numerically, whilst staying in touch with the data at all stages of modelling. In the present talk we explore the synthesis of both these paradigms, applied to modelling classical many-body systems. We propose a data-driven physics-informed inference framework for Helmholtz free energy functionals of such systems. Our approach is fully Bayesian and yields uncertainty quantification of the inferred model about its own predictions. The proposed algorithm trains humanly interpretable analytic free energy functionals using particle data, obtained from small-scale simulations. We focus on classical statistical-mechanical systems with excluded volume repulsive interactions and use a prototypical case of a one-dimensional fluid for algorithm validation. We are able to train canonical and grand-canonical representations of the underlying system. Extensions to higher-dimensional systems are conceptually straightforward. Using standard coarse-graining techniques, our results can also be made applicable to fluids with attractive-repulsive interactions.