Due to the widespread availability of effective antiretroviral therapy (ART) regimens, average lifespans of persons with HIV (PWH) in the United States have increased significantly in recent decades. In turn, the demographic profile of PWH has shifted. Older persons comprise an ever-increasing percentage of PWH, with this percentage expected to further increase in the coming years. This has profound implications for HIV treatment and care, as significant resources are required not only to manage HIV itself, but associated age-related comorbidities and health conditions that occur in aging PWH. Effective management of these challenges in the coming years requires accurate modeling of the PWH age structure. In the present work, we introduce several novel mathematical approaches related to this problem. We present a workflow combining a PDE model for the PWH population age structure, into which publicly-available HIV surveillance data is assimilated using the Ensemble Kalman Inversion (EKI) algorithm. This procedure allows us to rigorously reconstruct the age-dependent mortality trends for PWH over the last several decades. To project future trends, we introduce and analyze a novel variant of the Dynamic Mode Decomposition (DMD), nonnegative DMD. We show that nonnegative DMD provides physically-consistent projections of mortality and HIV diagnosis while remaining purely data-driven, and not requiring additional assumptions. We then combine these elements to provide forecasts for future trends in PWDH mortality and demographic evolution in the coming years.