Inverse problems, systems physiology, and human health: an interdisciplinary survey of three foundational problems

We would like to use data collected from humans to forward our understanding of physiological mechanics and to improve human health. However, because human data are costly and invasive, measurements are minimized, leaving us to attempt understanding a complex, multi-scale problem with sparse data.  In this talk I will demonstrate how the mathematical and computational problems in this context generalize and differ across specific physiological settings by discussing model estimation and forecasting in three contexts, the glucose-insulin system, the lung-ventilator system, and the neurovascular system. These cases will present new mathematical, computational, and scientific problems whose solutions and understanding requires an intradisciplinary understanding within mathematics combining knowledge from dynamical systems, bifurcation theory, approximation theory, machine learning, data assimilation, and mathematical physiology. Additionally, I will demonstrate how and why solutions to these broader problems that could potentially lead to positive impacts on health requires a highly interdisciplinary approach. Finally, I will present a mix of partial solutions and open problems related to the three above contexts.