  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% 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 catalogs 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 $\mu_{\text{LMC},\text{CEP}} =
18.688 \pm 0.093$, $\mu_{\text{LMC},\text{RRL}} = 18.52 \pm 0.14$, and
$\mu_{\text{SMC},\text{RRL}} = 18.88 \pm 0.17$ mag.
