  Characterizing the interior structure of exoplanets is essential for
understanding their diversity, formation, and evolution. As the interior of
exoplanets is inaccessible to observations, an inverse problem must be solved,
where numerical structure models need to conform to observable parameters such
as mass and radius. This is a highly degenerate problem whose solution often
relies on computationally-expensive and time-consuming inference methods such
as Markov Chain Monte Carlo.
  We present ExoMDN, a machine-learning model for the interior characterization
of exoplanets based on Mixture Density Networks (MDN). The model is trained on
a large dataset of more than 5.6 million synthetic planets below 25 Earth
masses consisting of an iron core, a silicate mantle, a water and high-pressure
ice layer, and a H/He atmosphere. We employ log-ratio transformations to
convert the interior structure data into a form that the MDN can easily handle.
Given mass, radius, and equilibrium temperature, we show that ExoMDN can
deliver a full posterior distribution of mass fractions and thicknesses of each
planetary layer in under a second on a standard Intel i5 CPU. Observational
uncertainties can be easily accounted for through repeated predictions from
within the uncertainties. We use ExoMDN to characterize the interior of 22
confirmed exoplanets with mass and radius uncertainties below 10% and 5%
respectively, including the well studied GJ 1214 b, GJ 486 b, and the
TRAPPIST-1 planets. We discuss the inclusion of the fluid Love number $k_2$ as
an additional (potential) observable, showing how it can significantly reduce
the degeneracy of interior structures. Utilizing the fast predictions of
ExoMDN, we show that measuring $k_2$ with an accuracy of 10% can constrain the
thickness of core and mantle of an Earth analog to $\approx13\%$ of the true
values.
