  We develop the neural network based "learning from regression uncertainty"
approach for automated detection of phases of matter in nonequilibrium active
systems. Taking the flocking phase transition of self-propelled active
particles described by the Vicsek model for example, we find that after
training a neural network for solving the inverse statistical problem, i.e.,
for performing the regression task of reconstructing the noise level from given
samples of such a nonequilibrium many-body complex system's steady state
configurations, the uncertainty of regression results obtained by the
well-trained network can actually be utilized to reveal possible phase
transitions in the system under study. The noise level dependence of regression
uncertainty assumes a non-trivial M-shape, and its valley appears at the
critical point of the flocking phase transition. By directly comparing this
regression-based approach with the widely-used classification-based "learning
by confusion" and "learning with blanking" approaches, we show that our
approach has practical effectiveness, efficiency, good generality for various
physical systems across interdisciplinary fields, and a greater possibility of
being interpretable via conventional notions of physics. These approaches can
complement each other to serve as a promising generic toolbox for investigating
rich critical phenomena and providing data-driven evidence on the existence of
various phase transitions, especially for those complex scenarios associated
with first-order phase transitions or nonequilibrium active systems where
traditional research methods in physics could face difficulties.
