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Reaction–diffusion equations are commonly used to model a diverse array of complex systems, including biological, chemical, and physical processes. Typically, these models are phenomenological, requiring the fitting of parameters to experimental data. In the present work, we introduce a novel formalism to construct reaction–diffusion models that is grounded in the principle of maximum entropy. This new formalism aims to incorporate various types of experimental data, including ensemble currents, distributions at different points in time, or moments of such. To this end, we expand the framework of Schrödinger bridges and maximum caliber problems to nonlinear interacting systems. We illustrate the usefulness of the proposed approach by modeling the evolution of (i) a morphogen across the fin of a zebrafish and (ii) the population of two varieties of toads in Poland, so as to match the experimental data. 

Evanescent random walks are instances of stochastic processes that terminate at a specific rate. They have proved relevant in modeling diverse behaviors of complex systems from protein degra- dation in gene networks (Ali and Brewster (2022), Ghusinga, et al. (2017), and Ham, et al. (2024)) and “nonprocessive” motor proteins (Kolomeisky and Fisher (2007)) to decay of diffusive radioactive matter (Zoia (2008)). The present work aims to extend a well-established estimation and control problem, the so-called Schr¨odinger’s bridge problem, to evanescent diffusion processes. Specifically, the authors seek the most likely law on the path space that restores consistency with two marginal densities — One is the initial probability density of the flow, and the other is a density of killed particles. The Schr¨odinger’s bridge problem can be interpreted as an estimation problem but also as a control problem to steer the stochastic particles so as to match specified marginals. The focus of previous work in Eldesoukey, et al. (2024) has been to tackle Schr¨odinger’s problem involving a constraint on the spatio-temporal density of killed particles, which the authors revisit here. The authors then expand on two related problems that instead separately constrain the temporal and the spatial marginal densities of killed particles. The authors derive corresponding Schr¨odinger systems that contain coupled partial differential equa- tions that solve such problems. The authors also discuss Fortet-Sinkhorn-like algorithms that can be used to construct the sought bridges numerically.