  The kernel ridge regression (KRR) method with Gaussian kernel is used to
improve the description of the nuclear charge radius by several
phenomenological formulae. The widely used $A^{1/3}$, $N^{1/3}$ and $Z^{1/3}$
formulae, and their improved versions by considering the isospin dependence are
adopted as examples. The parameters in these six formulae are refitted using
the Levenberg-Marquardt method, which give better results than the previous
ones. The radius for each nucleus is predicted with the KRR network, which is
trained with the deviations between experimental and calculated nuclear charge
radii. For each formula, the resultant root-mean-square deviations of 884
nuclei with proton number $Z \geq 8$ and neutron number $N \geq 8$ can be
reduced to about 0.017~fm after considering the modification of the KRR method.
The extrapolation ability of the KRR method for the neutron-rich region is
examined carefully and compared with the radial basis function method. It is
found that the improved nuclear charge radius formulae by KRR method can avoid
the risk of overfitting and have a good extrapolation ability. The influence of
the ridge penalty term on the extrapolation ability of the KRR method is also
discussed. At last, the nuclear charge radii of several recently observed K and
Ca isotopes have been analyzed.
