Unbalanced Diffusion Schrödinger Bridge
This work addresses a domain-specific problem for modeling natural processes with changing population sizes, such as in biology and chemistry, and is incremental as it extends existing methods to handle mass changes.
The paper tackled the limitation of existing neural Schrödinger bridges that assume conservation of mass by introducing unbalanced diffusion Schrödinger bridges to model temporal evolution with arbitrary finite mass, achieving applications in predicting molecular responses to cancer drugs and simulating viral variant spread.
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 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.