Partially observed Schrödinger flows: an application to stochastic thermodynamics*

Schrödinger bridges have emerged as an enabling theory for unveiling the stochastic dynamics of systems based on marginal observations at different points in time. The terminology "bridge'' refers to a probability law that suitably interpolates such marginals. The theory plays a pivotal role in a variety of contemporary developments in machine learning, stochastic control, thermodynamics, and biology, to name a few, impacting disciplines such as single-cell genomics, meteorology, and robotics. In this talk, we generalize Schrödinger's paradigm of bridges to account for partial observations. In doing so, we propose a framework that seeks the most likely underlying stochastic dynamics that match our limited observations. At the same time, the framework enables the estimation of quantities of interest, such as entropy production, heat, or work over a finite-time transition, when we only have access to partial information. This limited information can encompass a variety of data types, ranging from knowledge of distribution moments at different time points to knowledge of the average of functions on paths (i.e. averaged currents). We illustrate the practical applicability of the framework, given only knowledge of some averaged current over the transition of an unknown Markovian thermodynamic system, by finding the most likely underlying dynamics that led to those measurements, as well as estimating the entropy production incurred by the system along that thermodynamic transition.