Data-driven dynamical modeling is an emerging and powerful tool for analyzing, predicting, and controlling complex systems in engineering and physical sciences. Traditionally, modeling of dynamical systems uses mathematical approaches like differential equations;​ modern approaches leverage advances in machine learning to discovery models directly from system measurements. In engineering and physical sciences, first-principles and physics-based models are ubiquitous, but only allow for the modeling of dynamics with a limited accuracy;​ purely data-driven models of dynamical systems are able to learn complex relationships, but can become unconstrained without leveraging known physics. As data-driven dynamical modeling continues to gain momentum, it is imperative that researchers utilize a hybrid modeling approach to combine domain knowledge and measurements to model complex systems.

Discrepancy modeling for dynamical systems is a hybrid modeling approach that aims to resolve the mismatch between model estimations and measurement data due to missing physics. In this dissertation, we focus on shifting the view of discrepancies as ’errors’ or ’residuals’ to highly valuable measures for model improvement and scientific insight. We discuss two nuanced, yet distinct, discrepancy modeling approaches for estimating missing physics and demonstrate how different model discovery methods can be used interchangeably within this framework. Further, we emphasize discrepancy modeling considerations and trade-offs related to model interpretability and sensor constraints. The field of data-driven engineering for dynamical systems has an opportunity to improve system characterization and provide scientific insight by disambiguating deterministic and random effects within the model-measurement mismatch.

One such complex system is the human musculoskeletal system, and modeling the musculoskeletal system is of great interest to the biomechanics community. Great strides in clinical insight have emerged from modeling and simulation using physics-based and physiologically-detailed models. However, the complexity of the human body is challenging to represent as a musculoskeletal simulation, especially for clinical populations, limiting utility of such techniques. Machine learning has been employed to study human movement, but historically focuses on feature extraction and classification. In viewing the human body as a dynamical system, data-driven dynamical modeling techniques can be employed to tackle heterogeneous, nonlinear, and complex challenges in treating pathology, enhancing mobility, and personalizing rehabilitation.

In demonstrating data-driven dynamical modeling for human movement, we focus on two major challenges:​ (1) prescribing assistive devices and (2) collecting rich datasets of movement for health monitoring. In this dissertation, we apply discrepancy modeling to characterize individual responses to passive-elastic ankle exoskeletons during walking. A neural network-based discrepancy model successfully quantified complex changes in gait kinematics and electromyography during exoskeleton walking;​ yet, kinematics and electromyography alone were insufficient to fully capture muscle-level changes in responses to ankle exoskeletons. Understanding how an individual will respond to an assistive device and what data are needed to encode responses can help optimize the prescription process and uncover underlying mechanisms driving gait changes. The final study in this dissertation combined deep learning, time-delay embedding, and sparse sensing for full-state reconstruction of complex systems, including for personalized human movement tracking. For example, we can expand biomechanical datasets by mapping from as few as a single sensor to a full dataset of biomechanical states. This work will enable robust motion tracking in-the-wild for monitoring and supporting movement-based health outcomes;​ more generally, this mathematical architecture will enable state estimation of dynamical systems with mobile sensors without the challenge of sensor path planning.

Using data to discover dynamical models is transforming how we study complex systems. The work presented in this dissertation demonstrates how we can build models and extract insight from time-series data. While methodologically new to the field of biomechanics, modern advances in data-driven dynamical methods are a powerful tool for quantifying and understanding musculoskeletal biomechanics – and beyond. This dissertation provides the foundation for developing and deploying machine learning for dynamical models of human movement.