A proximal approach to the Schr\"odinger bridge problem with incomplete information and application to contamination tracking in water networks

Abstract

In this work, we study a discrete Schr\"odinger bridge problem with partial marginal observations. A main difficulty compared to the classical Schr\"odinger bridge formulation is that our problem is not strictly convex and standard Sinkhorn-type methods cannot be directly applied. To address this issue, we propose a scalable computational method based on an entropic proximal scheme. Furthermore, we develop a framework for this problem that includes duality results, characterization of the optimal solutions, and an observability condition that determines when the optimal solution is unique. We validate the method on the problem of estimating contamination in a water distribution network, where the partial marginals correspond to measured pollutant concentrations at the sensor locations. The experiments were conducted on a laboratory-scale water distribution network.