  Holography relates gravitational theories in five dimensions to
four-dimensional quantum field theories in flat space. Under this map, the
equation of state of the field theory is encoded in the black hole solutions of
the gravitational theory. Solving the five-dimensional Einstein's equations to
determine the equation of state is an algorithmic, direct problem. Determining
the gravitational theory that gives rise to a prescribed equation of state is a
much more challenging, inverse problem. We present a novel approach to solve
this problem based on physics-informed neural networks. The resulting algorithm
is not only data-driven but also informed by the physics of the Einstein's
equations. We successfully apply it to theories with crossovers, first- and
second-order phase transitions.
