Unbalanced Diffusion Schrödinger Bridge

Abstract

_Schrödinger 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 Schrödinger 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_ (SDEs) 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.