  We consider a non-linear heat equation $\partial_t u = \Delta u +
B(u,Du)+P(u)$ posed on the $d$-dimensional torus, where $P$ is a polynomial of
degree at most $3$ and $B$ is a bilinear map that is not a total derivative. We
show that, if the initial condition $u_0$ is taken from a sequence of smooth
Gaussian fields with a specified covariance, then $u$ exhibits norm inflation
with high probability. A consequence of this result is that there exists no
Banach space of distributions which carries the Gaussian free field on the 3D
torus and to which the DeTurck-Yang-Mills heat flow extends continuously, which
complements recent well-posedness results in arXiv:2111.10652 and
arXiv:2201.03487. Another consequence is that the (deterministic) non-linear
heat equation exhibits norm inflation, and is thus locally ill-posed, at every
point in the Besov space $B^{-1/2}_{\infty,\infty}$; the space
$B^{-1/2}_{\infty,\infty}$ is an endpoint since the equation is locally
well-posed for $B^{\eta}_{\infty,\infty}$ for every $\eta>-\frac12$.
