  We compute the nucleon axial and pseudoscalar form factors using three
$N_f=$2+1+1 twisted mass fermion ensembles with all quark masses tuned to
approximately their physical values. The values of the lattice spacings of
these three physical point ensembles are 0.080 fm, 0.068 fm, and 0.057 fm, and
spatial sizes 5.1 fm, 5.44 fm, and 5.47 fm, respectively, yielding $m_\pi
L$>3.6. Convergence to the ground state matrix elements is assessed using
multi-state fits. We study the momentum dependence of the three form factors
and check the partially conserved axial-vector current (PCAC) hypothesis and
the pion pole dominance (PPD). We show that in the continuum limit, the PCAC
and PPD relations are satisfied. We also show that the Goldberger-Treimann
relation is approximately fulfilled and determine the Goldberger-Treiman
discrepancy. We find for the nucleon axial charge $g_A$=1.245(28)(14), for the
axial radius $\langle r^2_A \rangle$=0.339(48)(06) fm$^2$, for the pion-nucleon
coupling constant $g_{\pi NN} \equiv \lim_{Q^2 \rightarrow -m_\pi^2} G_{\pi
NN}(Q^2)$=13.25(67)(69) and for $G_P(0.88m_{\mu}^2)\equiv g_P^*$=8.99(39)(49).
