We model the SEDs of 23 protoplanetary disks in the Taurus-Auriga star-forming region using detailed disk models and a Bayesian approach. This is made possible by combining these models with artificial neural networks to drastically speed up their performance. Such a setup allows us to confront $\alpha$-disk models with observations while accounting for several uncertainties and degeneracies. Our results yield high viscosities for many sources, which is not consistent with recent measurements of low turbulence levels in disks. This inconsistency could imply that viscosity is not the main mechanism for angular momentum transport in disks, and that alternatives such as disk winds play an important role in this process. We also find that our SED-derived disk masses are systematically higher (median difference of a factor of $\sim$3 higher) than those obtained solely from (sub)mm fluxes, suggesting that part of the disk emission could still be optically thick at (sub)mm wavelengths. This effect is particularly relevant for disk population studies and alleviates observational tensions between the masses of protoplanetary disks and exoplanetary systems.