Deep learning probability flows and entropy production rates in active
  matter
Nicholas M. Boffi Eric Vanden-Eijnden
Active matter systems, from self-propelled colloids to motile bacteria, are
characterized by the conversion of free energy into useful work at the
microscopic scale. These systems generically involve physics beyond the reach
of equilibrium statistical mechanics, and a persistent challenge has been to
understand the nature of their nonequilibrium states. The entropy production
rate and the magnitude of the steady-state probability current provide
quantitative ways to do so by measuring the breakdown of time-reversal symmetry
and the strength of nonequilibrium transport of measure. Yet, their efficient
computation has remained elusive, as they depend on the system's unknown and
high-dimensional probability density. Here, building upon recent advances in
generative modeling, we develop a deep learning framework that estimates the
score of this density. We show that the score, together with the microscopic
equations of motion, gives direct access to the entropy production rate, the
probability current, and their decomposition into local contributions from
individual particles, spatial regions, and degrees of freedom. To represent the
score, we introduce a novel, spatially-local transformer-based network
architecture that learns high-order interactions between particles while
respecting their underlying permutation symmetry. We demonstrate the broad
utility and scalability of the method by applying it to several
high-dimensional systems of interacting active particles undergoing
motility-induced phase separation (MIPS). We show that a single instance of our
network trained on a system of 4096 particles at one packing fraction can
generalize to other regions of the phase diagram, including systems with as
many as 32768 particles. We use this observation to quantify the spatial
structure of the departure from equilibrium in MIPS as a function of the number
of particles and the packing fraction.