  We develop a novel data-driven approach to the inverse problem of classical
statistical mechanics: given experimental data on the collective motion of a
classical many-body system, how does one characterise the free energy landscape
of that system? By combining non-parametric Bayesian inference with
physically-motivated constraints, we develop an efficient learning algorithm
which automates the construction of approximate free energy functionals. In
contrast to optimisation-based machine learning approaches, which seek to
minimise a cost function, the central idea of the proposed Bayesian inference
is to propagate a set of prior assumptions through the model, derived from
physical principles. The experimental data is used to probabilistically weigh
the possible model predictions. This naturally leads to humanly interpretable
algorithms with full uncertainty quantification of predictions. In our case,
the output of the learning algorithm is a probability distribution over a
family of free energy functionals, consistent with the observed particle data.
We find that surprisingly small data samples contain sufficient information for
inferring highly accurate analytic expressions of the underlying free energy
functionals, making our algorithm highly data efficient. We consider excluded
volume particle interactions, which are ubiquitous in nature, whilst being
highly challenging for modelling in terms of free energy. To validate our
approach we consider the paradigmatic case of one-dimensional fluid and develop
inference algorithms for the canonical and grand-canonical
statistical-mechanical ensembles. Extensions to higher-dimensional systems are
conceptually straightforward, whilst standard coarse-graining techniques allow
one to easily incorporate attractive interactions.
