  We use the macroscopic fluctuation theory (MFT) to study large current
fluctuations in non-stationary diffusive lattice gases. We identify two
universality classes of these fluctuations which we call elliptic and
hyperbolic. They emerge in the limit when the deterministic mass flux is small
compared to the mass flux due to the shot noise. The two classes are determined
by the sign of compressibility of \emph{effective fluid}, obtained by mapping
the MFT into an inviscid hydrodynamics. An example of the elliptic class is the
Symmetric Simple Exclusion Process where, for some initial conditions, we can
solve the effective hydrodynamics exactly. This leads to a super-Gaussian
extreme current statistics conjectured by Derrida and Gerschenfeld (2009) and
yields the optimal path of the system. For models of the hyperbolic class the
deterministic mass flux cannot be neglected, leading to a different extreme
current statistics.
