  Physics-based and first-principles models pervade the engineering and
physical sciences, allowing for the ability to model the dynamics of complex
systems with a prescribed accuracy. The approximations used in deriving
governing equations often result in discrepancies between the model and
sensor-based measurements of the system, revealing the approximate nature of
the equations and/or the signal-to-noise ratio of the sensor itself. In modern
dynamical systems, such discrepancies between model and measurement can lead to
poor quantification, often undermining the ability to produce accurate and
precise control algorithms. We introduce a discrepancy modeling framework to
identify the missing physics and resolve the model-measurement mismatch with
two distinct approaches: (i) by learning a model for the evolution of
systematic state-space residual, and (ii) by discovering a model for the
deterministic dynamical error. Regardless of approach, a common suite of
data-driven model discovery methods can be used. The choice of method depends
on one's intent (e.g., mechanistic interpretability) for discrepancy modeling,
sensor measurement characteristics (e.g., quantity, quality, resolution), and
constraints imposed by practical applications (e.g., modeling approaches using
the suite of data-driven modeling methods on three continuous dynamical systems
under varying signal-to-noise ratios. Finally, we emphasize structural
shortcomings of each discrepancy modeling approach depending on error type. In
summary, if the true dynamics are unknown (i.e., an imperfect model), one
should learn a discrepancy model of the missing physics in the dynamical space.
Yet, if the true dynamics are known yet model-measurement mismatch still
exists, one should learn a discrepancy model in the state space.
