A case study with step-by-step code implementation

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11 hours ago

Differential equations function a robust framework to capture and understand the dynamic behaviors of physical systems. By describing how variables change in relation to one another, they supply insights into system dynamics and permit us to make predictions concerning the system’s future behavior.

Nonetheless, a typical challenge we face in lots of real-world systems is that their governing differential equations are sometimes only partially known, with the unknown points manifesting in several ways:

• The parameters of the differential equation are unknown. A working example is wind engineering, where the governing equations of fluid dynamics are well-established, however the coefficients regarding turbulent flow are highly uncertain.
• The functional forms of the differential equations are unknown. For example, in chemical engineering, the precise functional type of the speed equations is probably not fully understood as a consequence of the uncertainties in rate-determining steps and response pathways.
• Each functional forms and parameters are unknown. A first-rate example is battery state modeling, where the commonly used equivalent circuit model only partially captures the current-voltage relationship (the functional type of the missing physics is subsequently unknown). Furthermore, the model itself incorporates unknown parameters (i.e., resistance and capacitance values).

Such partial knowledge of the governing differential equations hinders our understanding and control of those dynamical systems. Consequently, inferring these unknown components based on observed data becomes an important task in dynamical system modeling.

Broadly speaking, this means of using observational data to recuperate governing equations of dynamical systems falls within the domain of system identification. Once discovered, we will readily use these equations to predict future states of the system, inform control strategies for the systems, or…