Physics-Constrained Data-Driven Variational Method for Discrepancy Modeling

This paper presents a data-driven discrepancy modeling method that variationally embeds measured data in the modeling and analysis framework. The proposed method exploits the residual between the first-principles theory and sensor-based measurements from the dynamical system, and it augments the physics-based model with a variationally derived loss function that is comprised of this residual. The method was first developed in the context of linear elasticity (Masud and Goraya, J. Appl. Mech. 89 (11), 111001 (2022)) wherein the relation between the discrepancy model and loss terms was derived to show that the data embedding terms behave like residual-based least-squares regression functions. An interpretation of the stabilization tensor as a kernel function was formally established and its role in assimilating a-priori knowledge of the problem in the modeling method was highlighted. The present paper employs linear elastodynamics as a model problem where the Data-Driven Variational (DDV) method incorporates high-fidelity data into the forward simulations, thereby driving the problem with not only the boundary and initial conditions, but also by measurement data that is taken at only a small subset of the total domain. The effect of the loss function on the time-dependent response of the system is investigated under a variety of loading conditions and model discrepancies. The energy and Morlet wavelet analyses reveal that the problem with embedded data recovers the energy and the fundamental frequency band of the target system. Time histories of strain energy and kinetic energy of a cantilever beam undergoing damped oscillations are recovered by including known data in an undamped model to highlight the data-driven discrepancy modeling feature of the method under the combined effect of parameter and model discrepancy.

A machine learning approach to quantify individual gait responses to ankle exoskeletons

Predicting an individual's response to an exoskeleton and understanding what data are needed to characterize responses remains challenging. Specifically, we lack a theoretical framework capable of quantifying heterogeneous responses to exoskeleton interventions. We leverage a neural network-based discrepancy modeling framework to quantify complex changes in gait in response to passive ankle exoskeletons in nondisabled adults. Discrepancy modeling aims to resolve dynamical inconsistencies between model predictions and real-world measurements. Neural networks identified models of (i) Nominal gait, (ii) Exoskeleton (Exo) gait, and (iii) the Discrepancy (i.e., response) between them. If an Augmented (Nominal+Discrepancy) model captured exoskeleton responses, its predictions should account for comparable amounts of variance in Exo gait data as the Exo model. Discrepancy modeling successfully quantified individuals' exoskeleton responses without requiring knowledge about physiological structure or motor control: a model of Nominal gait augmented with a Discrepancy model of response accounted for significantly more variance in Exo gait (median R2 for kinematics (0.928-0.963) and electromyography (0.665-0.788), (p<0.042)) than the Nominal model (median R2 for kinematics (0.863-0.939) and electromyography (0.516-0.664)). However, additional measurement modalities and/or improved resolution are needed to characterize Exo gait, as the discrepancy may not comprehensively capture response due to unexplained variance in Exo gait (median R2 for kinematics (0.954-0.977) and electromyography (0.724-0.815)). These techniques can be used to accelerate the discovery of individual-specific mechanisms driving exoskeleton responses, thus enabling personalized rehabilitation.