Discovering Differential Equations with Physics-Informed Neural Networks and Symbolic Regression | by Shuai Guo | Jul, 2023

A case study with step-by-step code implementation

Photo by Steven Coffey on Unsplash

Differential equations serve as a powerful framework to capture and understand the dynamic behaviors of physical systems. By describing how variables change in relation to each other, they provide insights into system dynamics and allow us to make predictions about the system’s future behavior.

However, a common challenge we face in many real-world systems is that their governing differential equations are often only partially known, with the unknown aspects manifesting in several ways:

  • The parameters of the differential equation are unknown. A case in point is wind engineering, where the governing equations of fluid dynamics are well-established, but the coefficients relating to turbulent flow are highly uncertain.
  • The functional forms of the differential equations are unknown. For instance, in chemical engineering, the exact functional form of the rate equations may not be fully understood due to the uncertainties in rate-determining steps and reaction pathways.
  • Both functional forms and parameters are unknown. A prime example is battery state modeling, where the commonly used equivalent circuit model only partially captures the current-voltage relationship (the functional form of the missing physics is therefore unknown). Moreover, the model itself contains unknown parameters (i.e., resistance and capacitance values).
Figure 1. The governing equations of many real-world dynamical systems are only partially known. (Image by this blog author)

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

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

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