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.

该研究提出一种新的神经参数化方法：非平衡扩散 Schrödinger Bridge（DSBs）。该方法可以模拟当端点是概率分布，且质量未被守恒时，人口的时间演化，并可应用于分析癌症药物的单细胞响应和病毒变异的传播过程。