We have performed the first Bayesian neural-network analysis of neutrino-deuteron scattering data. The nucleon axial form factor has been extracted from quasielastic scattering data collected by the Argonne National Laboratory (ANL) bubble chamber experiment using a model-independent parametrization. The results are in agreement with previous determinations only when the low $0.05 < Q^2 < 0.10$~GeV$^2$ region is excluded from the analysis. This suggests that corrections from the deuteron structure may play a crucial role at low $Q^2$, although experimental errors in this kinematic region could have also been underestimated. With new and more precise measurements of neutrino-induced quasielastic scattering on hydrogen and deuterium, the present framework would be readily applicable to unravel the axial structure of the nucleon.