Unbalanced Diffusion Schr\"odinger Bridge

TL;DR

This paper introduces unbalanced diffusion Schr"odinger bridges that model population dynamics with changing mass, extending existing SB frameworks to more accurately reflect biological and physical processes involving birth and death events.

Contribution

It develops a new theoretical framework and scalable algorithms for unbalanced diffusion Schr"odinger bridges, allowing modeling of systems with variable total mass.

Findings

Effective in predicting cellular responses to cancer drugs

Simulates emergence and spread of viral variants

Provides scalable training algorithms for unbalanced SBs

Abstract

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.