  We present results for the form factors of the isovector axial vector current
in the nucleon state using large scale simulations of lattice QCD. The
calculations were done using eight ensembles of gauge configurations generated
by the MILC collaboration using the HISQ action with 2+1+1 dynamical flavors.
These ensembles span three lattice spacings $a \approx 0.06, 0.09$ and $0.12$
fm and light-quark masses corresponding to the pion masses $M_\pi \approx 135,
225$ and $310$ MeV. High-statistics estimates allow us to quantify systematic
uncertainties in the extraction of $G_A(Q^2)$ and the induced pseudoscalar form
factor $\tilde{G}_P(Q^2)$. We perform a simultaneous extrapolation in the
lattice spacing, lattice volume and light-quark masses of the axial charge
radius $r_A$ data to obtain physical estimates. Using the dipole ansatz to fit
the $Q^2$ behavior we obtain $r_A|_{\rm dipole} = 0.49(3)$ fm, which
corresponds to ${\cal M}_A = 1.39(9)$ GeV, and is consistent with ${\cal M}_A =
1.35(17)$ GeV obtained by the miniBooNE collaboration. The estimate obtained
using the $z$-expansion is $r_A|_{z-{\rm expansion}} = 0.46(6)$ fm, and the
combined result is $r_A|_{\rm combined} = 0.48(4)$ fm. Analysis of the induced
pseudoscalar form factor $\tilde{G}_P(Q^2)$ yields low estimates for $g_P^\ast$
and $g_{\pi {\rm NN}}$ compared to their phenomenological values. To understand
these, we analyze the partially conserved axial current (PCAC) relation by also
calculating the pseudoscalar form factor. We find that these low values are due
to large deviations in the PCAC relation between the three form factors and
from the pion-pole dominance hypothesis.
