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.

翻译：薛定谔桥（SBs）为物理、化学或生物系统中群体时间演化的建模提供了一个优雅的框架。由于新物种的出现或出生与死亡事件，此类自然过程通常伴随着群体规模随时间的变化。然而，现有的SBs神经参数化方法（如扩散薛定谔桥（DSB））仅限于随机过程端点均为概率测度且假设满足质量守恒约束的场景。为突破这一局限，我们提出了非平衡DSB，能够对具有任意有限质量的边缘分布的时间演化进行建模。通过推导带有死亡项与出生项的随机微分方程的时间反演，我们实现了这一目标。我们提出了两种新颖的算法方案，包含用于训练非平衡DSB的可扩展目标函数，并提供了理论分析，同时展示了其在预测不同癌症药物作用下异质性分子单细胞响应，以及模拟新病毒变种出现与传播等挑战性应用中的效果。