  The $O(3)$ non-linear sigma model (NLSM) is a prototypical field theory for
QCD and ferromagnetism, and provides a simple system in which to study
topological effects. In lattice QCD, the gradient flow has been demonstrated to
remove ultraviolet singularities from the topological susceptibility. In
contrast, lattice simulations of the NLSM find that the topological
susceptibility diverges in the continuum limit, even in the presence of the
gradient flow. We introduce a $\theta$-term and analyze the topological charge
as a function of $\theta$ under the gradient flow. Our results show that
divergence persists in the presence of the flow, even at non-zero $\theta$.
