  Being able to describe accurately the dynamics and steady-states of driven
and/or dissipative but quantum correlated lattice models is of fundamental
importance in many areas of science: from quantum information to biology. An
efficient numerical simulation of large open systems in two spatial dimensions
is a challenge. In this work, we develop a tensor network method, based on an
infinite Projected Entangled Pair Operator (iPEPO) ansatz, applicable directly
in the thermodynamic limit. We incorporate techniques of finding optimal
truncations of enlarged network bonds by optimising an objective function
appropriate for open systems. Comparisons with numerically exact calculations,
both for the dynamics and the steady-state, demonstrate the power of the
method. In particular, we consider dissipative transverse quantum Ising and
driven-dissipative hard core boson models in non-mean field limits, proving
able to capture substantial entanglement in the presence of dissipation. Our
method enables to study regimes which are accessible to current experiments but
lie well beyond the applicability of existing techniques.
