NSF-BSF: Learning from Ensemble Data: trajectory recovery and flow-map inference for multiple systems

Understanding how complex systems evolve over time, from evolving opinions on social networks to the interactions of cells or genes, relies on reconstructing their underlying dynamics from observational data. Many experiments and measurements, however, only provide snapshots of system ensembles rather than complete trajectories, posing a significant challenge for data-driven modeling. This project will develop new mathematical and computational tools to learn the rules governing these high-dimensional systems directly from ensemble data. By doing so, the research will advance scientific discoveries using ensemble data and enable accurate predictions in fields ranging from medical imaging and biological dynamics to opinion dynamics modeling. The methods and software produced will be openly shared, providing hands-on training for undergraduate and graduate students, and establishing an international collaborative network that advances scientific knowledge and trains future researchers. The project will introduce new theoretical and computational tools for learning dynamics from ensemble data. The research uses weak-form partial differential equations and gradient flows to enable learning from the empirical distributions of the ensemble data. It targets both single-system and cross-system learning by developing an automatic kernel regression and a task-specific attentional model. The research will establish mathematical foundations for both single-system and multi-system learning through theoretical analysis of identifiability and convergence rates. The resulting algorithms will be implemented in open-source software, and applied to exemplar problems in imaging and interacting particle systems. This work will advance mathematical theory and provide robust computational tools for data-driven scientific discovery. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.