A back-propagation neural network is utilized to fit the potential energy surfaces of the ${\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$ ion, using the ab initio data points of Dykstra and Swope, and the Meyer, Botschwina, and Burton ab initio data points. We used the standard back-propagation formulation and have also proposed a symmetric formulation to account for the symmetry of the ${\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$ molecule. To test the quality of the fits we computed the vibrational levels using the correlation function quantum Monte Carlo method. We have compared our results with the available experimental results and with results obtained using other potential energy surfaces. The vibrational levels are in very good agreement with the experiment and the back-propagation fitting is of the same quality of the available potential energy surfaces.