  The recent release of the second Gravitational-Wave Transient Catalog
(GWTC-2) has increased significantly the number of known GW events, enabling
unprecedented constraints on formation models of compact binaries. One pressing
question is to understand the fraction of binaries originating from different
formation channels, such as isolated field formation versus dynamical formation
in dense stellar clusters. In this paper, we combine the $\texttt{COSMIC}$
binary population synthesis suite and the $\texttt{CMC}$ code for globular
cluster evolution to create a mixture model for black hole binary formation
under both formation scenarios. For the first time, these code bodies are
combined self-consistently, with $\texttt{CMC}$ itself employing
$\texttt{COSMIC}$ to track stellar evolution. We then use a deep-learning
enhanced hierarchical Bayesian analysis to constrain the mixture fraction $f$
between formation models, while simultaneously constraining the common envelope
efficiency $\alpha$ assumed in $\texttt{COSMIC}$ and the initial cluster virial
radius $r_v$ assumed in $\texttt{CMC}$. Under specific assumptions about other
uncertain aspects of isolated binary and globular cluster evolution, we report
the median and $90\%$ confidence interval of three physical parameters
$(f,\alpha,r_v)=(0.20^{+0.32}_{-0.18},2.26^{+2.65}_{-1.84},2.71^{+0.83}_{-1.17})$.
This simultaneous constraint agrees with observed properties of globular
clusters in the Milky Way and is an important first step in the pathway toward
learning astrophysics of compact binary formation from GW observations.
