  Comparing unpaired samples of a distribution or population taken at different
points in time is a fundamental task in many application domains where
measuring populations is destructive and cannot be done repeatedly on the same
sample, such as in single-cell biology. Optimal transport (OT) can solve this
challenge by learning an optimal coupling of samples across distributions from
unpaired data. However, the usual formulation of OT assumes conservation of
mass, which is violated in unbalanced scenarios in which the population size
changes (e.g., cell proliferation or death) between measurements. In this work,
we introduce NubOT, a neural unbalanced OT formulation that relies on the
formalism of semi-couplings to account for creation and destruction of mass. To
estimate such semi-couplings and generalize out-of-sample, we derive an
efficient parameterization based on neural optimal transport maps and propose a
novel algorithmic scheme through a cycle-consistent training procedure. We
apply our method to the challenging task of forecasting heterogeneous responses
of multiple cancer cell lines to various drugs, where we observe that by
accurately modeling cell proliferation and death, our method yields notable
improvements over previous neural optimal transport methods.
