We propose a novel tensor-network method to achieve accurate and efficient simulations of Abelian lattice gauge theories (LGTs) in $(2+1)\mathrm{D}$ for both ground-state and real-time dynamics. The first key is to identify a gauge canonical form of gauge-invariant tensor-network states, which already simplifies existing algorithms for $(1+1)\mathrm{D}$ LGTs. The second key is to employ the gauge canonical form of a projected entangled-pair state (PEPS) combining with variational Monte Carlo, enabling efficient computations for $(2+1)\mathrm{D}$ LGTs. We demonstrate the versatile capability of this approach for accurate ground-state simulation of pure ${\mathbb{Z}}_2$, ${\mathbb{Z}}_3$, and ${\mathbb{Z}}_4$ gauge theory, odd-${\mathbb{Z}}_2$ gauge theories, and ${\mathbb{Z}}_2$ gauge theory coupled to hard-core bosons, on square lattices up to $32\times 32$. Furthermore, we demonstrate that it allows for accurate simulations of real-time dynamics up to long time, exemplified by the dynamics of elementary excitations of the deconfined ${\mathbb{Z}}_2$ gauge field at $10\times 10$. This is also the first example of simulating the real-time dynamics of PEPS with variational Monte Carlo, whose impact may extend beyond gauge theory. Our Letter establishes gauge-invariant PEPS as a powerful approach for both ground-state and dynamical simulations, opening up a new avenue for nonperturbatively studying $(2+1)\mathrm{D}$ LGTs.

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• Phase transitions
• Lattice gauge theory
• Tensor network methods