The period of pulsation and the structure of the light curve for Cepheid and RR Lyrae variables depend on the fundamental parameters of the star: mass, radius, luminosity, and effective temperature. Here, we train artificial neural networks on theoretical pulsation models to predict the fundamental parameters of these stars based on their period and light-curve structure. We find significant improvements to estimates of these parameters made using light-curve structure and period over estimates made using only the period. Given that the models are able to reproduce most observables, we find that the fundamental parameters of these stars can be estimated up to 60 per cent more accurately when light-curve structure is taken into consideration. We quantify which aspects of light-curve structure are most important in determining fundamental parameters, and find, for example, that the second Fourier amplitude component of RR Lyrae light curves is even more important than period in determining the effective temperature of the star. We apply this analysis to observations of hundreds Cepheids in the Large Magellanic Cloud and thousands of RR Lyrae in the Magellanic Clouds and Galactic bulge to produce catalogues of estimated masses, radii, luminosities, and other parameters of these stars. As an example application, we estimate Wesenheit indices and use those to derive distance moduli to the Magellanic Clouds of {mu}_LMC,CEP_=18.688+/-0.093, {mu}_LMC,RRL_=18.52+/-0.14, and {mu}_SMC,RRL_=18.88+/-0.17mag.