  We show that the recently introduced ModMax theory of electrodynamics and its
Born-Infeld-like generalization are related by a flow equation driven by a
quadratic combination of stress-energy tensors. The operator associated to this
flow is a $4d$ analogue of the $T\bar{T}$ deformation in two dimensions. This
result generalizes the observation that the ordinary Born-Infeld Lagrangian is
related to the free Maxwell theory by a current-squared flow. As in that case,
we show that no analogous relationship holds in any other dimension besides
$d=4$. We also demonstrate that the $\mathcal{N}=1$ supersymmetric version of
the ModMax-Born-Infeld theory obeys a related supercurrent-squared flow which
is formulated directly in $\mathcal{N}=1$ superspace.
