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.

利用混合密度网络（MDN）和机器学习建立了 ExoMDN 模型，可以在 1 秒内处理获得行星每个层次的质量分数和厚度的完整后验分布，对小于 25 个地球质量的合成行星进行训练，并用于表征 22 个已确认的系外行星，包括 GJ 1214 b，GJ 486 b 和 TRAPPIST-1 行星等。模型可以方便地处理观察误差，并展示如何将流体 Love 数 k2 作为额外的（潜在的）观察量来减少内部结构的退化。利用 ExoMDN 的快速预测，测量约 10% 的 k2 精度可以约束地球类似体的核和地幔的厚度约为真实值的 13％。