  Nonequilibrium phase transitions are notably difficult to analyze because
their mechanisms depend on the system's dynamics in a complex way due to the
lack of time-reversal symmetry. To complicate matters, the system's
steady-state distribution is unknown in general. Here, the phase diagram of the
active Model B is computed with a deep neural network implementation of the
geometric minimum action method (gMAM). This approach unveils the
unconventional reaction paths and nucleation mechanism in dimensions 1, 2 and
3, by which the system switches between the homogeneous and inhomogeneous
phases in the binodal region. Our main findings are: (i) the mean time to
escape the phase-separated state is (exponentially) extensive in the system
size $L$, but it increases non-monotonically with $L$ in dimension 1; (ii) the
mean time to escape the homogeneous state is always finite, in line with the
recent work of Cates and Nardini~[Phys. Rev. Lett. 130, 098203]; (iii) at fixed
$L$, the active term increases the stability of the homogeneous phase,
eventually destroying the phase separation in the binodal for large but finite
systems. Our results are particularly relevant for active matter systems in
which the number of constituents hardly goes beyond $10^7$ and where
finite-size effects matter.
