  Monte Carlo methods have led to profound insights into the strong-coupling
behaviour of lattice gauge theories and produced remarkable results such as
first-principles computations of hadron masses. Despite tremendous progress
over the last four decades, fundamental challenges such as the sign problem and
the inability to simulate real-time dynamics remain. Neural network quantum
states have emerged as an alternative method that seeks to overcome these
challenges. In this work, we use gauge-invariant neural network quantum states
to accurately compute the ground state of $\mathbb{Z}_N$ lattice gauge theories
in $2+1$ dimensions. Using transfer learning, we study the distinct topological
phases and the confinement phase transition of these theories. For
$\mathbb{Z}_2$, we identify a continuous transition and compute critical
exponents, finding excellent agreement with existing numerics for the expected
Ising universality class. In the $\mathbb{Z}_3$ case, we observe a weakly
first-order transition and identify the critical coupling. Our findings suggest
that neural network quantum states are a promising method for precise studies
of lattice gauge theory.
