  We derive a complete classification of Floquet phases of interacting bosons
and fermions with $U(1)$ symmetry in two spatial dimensions. According to our
classification, there is a one-to-one correspondence between these Floquet
phases and rational functions $\pi(z) = a(z)/b(z)$ where $a(z)$ and $b(z)$ are
polynomials obeying certain conditions and $z$ is a formal parameter. The
physical meaning of $\pi(z)$ involves the stroboscopic edge dynamics of the
corresponding Floquet system: in the case of bosonic systems, $\pi(z) =
\frac{p}{q} \cdot \tilde{\pi}(z)$ where $\frac{p}{q}$ is a rational number
which characterizes the flow of quantum information at the edge during each
driving period, and $\tilde{\pi}(z)$ is a rational function which characterizes
the flow of $U(1)$ charge at the edge. A similar decomposition exists in the
fermionic case. We also show that $\tilde{\pi}(z)$ is directly related to the
time-averaged $U(1)$ current that flows in a particular geometry. This $U(1)$
current is a generalization of the quantized current and quantized
magnetization density found in previous studies of non-interacting fermionic
Floquet phases.
