Smoothly-Stochastic Dynamics: Estimation and Inference for Differential Equation Models

Modern sensors — including motion capture systems, remote sensing platforms, wearable devices, and infrastructure monitors — collect high-frequency measurements of processes that evolve over time. Making full use of this data requires mathematical models that describe how these systems change, along with statistical methods to fit those models reliably. This project advances the national interest by developing such tools with broad applicability across the health sciences, engineering, and ecological monitoring. This proposal's application focus is on human movement biomechanics, where understanding how people control and stabilize their motion can lead to improved injury rehabilitation, better assessment of movement disorders, and enhanced athletic training. The project will produce open-source software, tutorial publications, and training workshops to make these methods accessible to researchers across disciplines. It will support graduate student training and foster international research collaboration in statistics, biomechanics, and sports science. This project develops a statistical framework for smoothly-stochastic differential equation models, in which deterministic dynamics are augmented by smooth random forcing. Standard methods for fitting differential equation models to data treat discrepancies between model and observations as artifacts of data smoothing, but real systems exhibit persistent stochastic perturbations from unmodeled influences and model misspecification. The investigators will develop bias-corrected estimation procedures and new score-based inference methods for both linear systems (through Principal Differential Analysis) and nonlinear ordinary differential equation models (through gradient and integral matching). The project establishes theoretical guarantees, including consistency and asymptotic normality under simultaneous in-fill and expanding-domain asymptotics, and develops block bootstrap methods for uncertainty quantification from single trajectories. Extensions address sparsely and indirectly observed systems through profiling methods, and non-smooth stochastic disturbances through adapted particle filtering for non-Markovian systems. Model selection tools based on sparsity penalties and stationarity diagnostics will be developed. All of these are closely related to machine learning and artificial intelligence, and the methods will be applied to motion tracking data in biomechanics, in collaboration with researchers at University College Cork and the University of Technology Sydney, to characterize movement stability and its relationship to injury and performance. 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. NSF Award ID: 2610276 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Giles Hooker | Institution: University of Pennsylvania, PHILADELPHIA, PA | Award Amount: $225,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2610276 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2610276.html