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A gray-box model is one that has a known structure (generally constrained to a strict subset of the class of models it is drawn from) but has unknown parameters. Such models typically embody or reflect the underlying physical or mechanistic understanding we have about the system, as well as structural features such as the delineation of subsystems and their interconnections. The unknown parameters in the gray-box model then become the focus of our system identification efforts. In a variety of application domains, ranging from biology and medicine to power systems, the gray-box models that practitioners accept - as plausible representations of the reality they deal with every day - have been built up over decades of study, and are large, detailed and complex. In addition to being difficult to simulate or compute or design with, a significant feature of these models is the uncertainty associated with many or most of the parameters in the model. The data that one collects from the associated system is rarely rich enough to allow reliable identification of all these parameters, yet there are good reasons to not be satisfied with direct black-box identification of a reduced-order model. The challenge then is to develop meaningful reduced-order gray-box models that reflect the detailed, hard-won knowledge one has about the system, while being better suited to identification and simulation and control design than the original large model. Practitioners generally seem to have an intuitive understanding of what aspects of the original model structure, and which variables and parameters, should be retained in a physically or mechanistically meaningful reduced-order model for whatever aspect of the system behavior they are dealing with at a particular time. Can we capture and perhaps improve on what they are doing when they develop their (often informal) reduced models?