  Modeling living systems at the collective scale can be very challenging
because the individual constituents can themselves be complex and the
respective interactions between the constituents are not fully understood. With
the advent of high throughput experiments and in the age of big data,
data-driven methods are on the rise to overcome these challenges. Although
machine-learning approaches can help quantify correlations between the various
players, they do not directly shed light on the underlying physical principles
of such systems. To directly illuminate the underlying physical principles, we
present a data-driven method for obtaining the single particle density, from
which physical quantities can be readily extracted, as well as for solving the
Boltzmann equation for active matter -- a leading candidate for quantifying
living systems. If the system is near a steady state, a field theory can be
inferred to subsequently make physical predictions about the system. The method
is first developed analytically for a scalar field and subsequently calibrated
using simulated data. The method is then applied to an experimental system of
particles actively-driven by a {\it Serratia marcescens} bacterial swarm and in
the presence of spatially-localized UV light. The analysis demonstrates that
the particles are in steady state before and some time after the UV light and
obey a Gaussian field theory with a spatially-varying "mass". We use our
data-driven method to obtain the chemical potential, pressure, and entropy of
the particulate system. Specifically, we demonstrate that the entropy of the
particles increases in response to the UV light. In response to the UV light,
there is net flow of the particles away from the UV light. We conclude with
discussing other potential applications of the method to demonstrate its
breadth.
