  Given a symplectic class $[\omega]$ on a four torus $T^4$ (or a $K3$
surface), a folklore problem in symplectic geometry is whether symplectic forms
in $[\omega]$ are isotropic to each other. We introduce a family of nonlinear
Hodge heat flows on compact symplectic four manifolds to approach this problem,
which is an adaption of nonlinear Hodge theory in symplectic geometry. As a
particular example, we study a conformal Hodge heat flow in detail. We prove a
stability result of the flow near an almost Kahler structure $(M, \omega, g)$.
We also prove that, if $|\nabla \log u|$ stays bounded along the flow, then the
flow exists for all time for any initial symplectic form $\rho\in [\omega]$ and
it converges to $\omega$ smoothly along the flow with uniform control, where
$u$ is the volume potential of $\rho$.
