  Optimal transport (OT) theory has reshaped the field of generative modeling:
Combined with neural networks, recent \textit{Neural OT} (N-OT) solvers use OT
as an inductive bias, to focus on ``thrifty'' mappings that minimize average
displacement costs. This core principle has fueled the successful application
of N-OT solvers to high-stakes scientific challenges, notably single-cell
genomics. N-OT solvers are, however, increasingly confronted with practical
challenges: while most N-OT solvers can handle squared-Euclidean costs, they
must be repurposed to handle more general costs; their reliance on
deterministic Monge maps as well as mass conservation constraints can easily go
awry in the presence of outliers; mapping points \textit{across} heterogeneous
spaces is out of their reach. While each of these challenges has been explored
independently, we propose a new framework that can handle, natively, all of
these needs. The \textit{generative entropic neural OT} (GENOT) framework
models the conditional distribution $\pi_\varepsilon(\*y|\*x)$ of an optimal
\textit{entropic} coupling $\pi_\varepsilon$, using conditional flow matching.
GENOT is generative, and can transport points \textit{across} spaces, guided by
sample-based, unbalanced solutions to the Gromov-Wasserstein problem, that can
use any cost. We showcase our approach on both synthetic and single-cell
datasets, using GENOT to model cell development, predict cellular responses,
and translate between data modalities.
