  Predicting the critical temperature $T_c$ of new superconductors is a
notoriously difficult task, even for electron-phonon paired superconductors for
which the theory is relatively well understood. Early attempts by McMillan and
Allen and Dynes to improve on the weak-coupling BCS formula led to closed-form
approximate relations between $T_c$ and various measures of the phonon spectrum
and the electron-phonon interaction appearing in Eliashberg theory. Here we
propose that these approaches can be improved with the use of machine learning
algorithms. As an initial test, we train a model for identifying
low-dimensional descriptors using the $T_c < 10$ K data tested by Allen and
Dynes, and show that a simple analytical expression thus obtained improves upon
the Allen-Dynes fit. Furthermore, the prediction for the recently discovered
high $T_c$ material H$_3$S at high pressure is quite reasonable. Interestingly,
$T_c$'s for more recently discovered superconducting systems with a more
two-dimensional electron-phonon coupling, which do not follow Allen and Dynes'
expression, also do not follow our analytic expression. Thus, this machine
learning approach appears to be a powerful method for highlighting the need for
a new descriptor beyond those used by Allen and Dynes to describe their set of
isotropic electron-phonon coupled superconductors. We argue that this machine
learning method, and its implied need for a descriptor characterizing Fermi
surface properties, represents a promising new approach to superconductor
materials discovery which may eventually replace the serendipitous discovery
paradigm begun by Kamerlingh Onnes.
