Unbalanced Diffusion Schrödinger Bridge
Matteo Pariset Ya-Ping Hsieh Charlotte Bunne Andreas Krause Valentin De Bortoli
Schr\"odinger bridges (SBs) provide an elegant framework for modeling the
temporal evolution of populations in physical, chemical, or biological systems.
Such natural processes are commonly subject to changes in population size over
time due to the emergence of new species or birth and death events. However,
existing neural parameterizations of SBs such as diffusion Schr\"odinger
bridges (DSBs) are restricted to settings in which the endpoints of the
stochastic process are both probability measures and assume conservation of
mass constraints. To address this limitation, we introduce unbalanced DSBs
which model the temporal evolution of marginals with arbitrary finite mass.
This is achieved by deriving the time reversal of stochastic differential
equations with killing and birth terms. We present two novel algorithmic
schemes that comprise a scalable objective function for training unbalanced
DSBs and provide a theoretical analysis alongside challenging applications on
predicting heterogeneous molecular single-cell responses to various cancer
drugs and simulating the emergence and spread of new viral variants.