  ABRIDGED. The analysis of spectral energy distributions (SEDs) of
protoplanetary disks to determine their physical properties is known to be
highly degenerate. Hence, a Bayesian analysis is required to obtain parameter
uncertainties and degeneracies. The challenge here is computational speed, as
one radiative transfer model requires a couple of minutes to compute. We
performed a Bayesian analysis for 30 well-known protoplanetary disks to
determine their physical disk properties, including uncertainties and
degeneracies. To circumvent the computational cost problem, we created neural
networks (NNs) to emulate the SED generation process. We created two sets of
radiative transfer disk models to train and test two NNs that predict SEDs for
continuous and discontinuous disks. A Bayesian analysis was then performed on
30 protoplanetary disks with SED data collected by the DIANA project to
determine the posterior distributions of all parameters. We ran this analysis
twice, (i) with old distances and additional parameter constraints as used in a
previous study, to compare results, and (ii) with updated distances and free
choice of parameters to obtain homogeneous and unbiased model parameters. We
evaluated the uncertainties in the determination of physical disk parameters
from SED analysis, and detected and quantified the strongest degeneracies. The
NNs are able to predict SEDs within 1ms with uncertainties of about 5% compared
to the true SEDs obtained by the radiative transfer code. We find parameter
values and uncertainties that are significantly different from previous values
obtained by $\chi^2$ fitting. Comparing the global evidence for continuous and
discontinuous disks, we find that 26 out of 30 objects are better described by
disks that have two distinct radial zones. Also, we created an interactive tool
that instantly returns the SED predicted by our NNs for any parameter
combination.
