Modeling Dynamical Systems With Neural ODE: A Hands-on Guide | by Shuai Guo | Jan, 2024


Concepts, case studies, step-by-step implementations

Shuai Guo
Towards Data Science
Image generated by DALL-E.

For many physical systems, properly modeling their dynamical behaviors lays the foundation for system understanding, forecasting, and control. Mathematically, ordinary differential equations (ODEs), which describe the rate of change of system states over time, are widely used to characterize the dynamical systems. Correspondingly, many analytical and numerical solvers have been developed that can effectively solve the ODEs, thus allowing us to portray the system state trajectory.

However, there is one condition for using ODE solvers: we need to have complete knowledge of the ODEs. For many real-world, complex systems, unfortunately, this requirement is rarely fulfilled: for some, we may know the functional form of the ODEs but don’t know the associated model parameters; for others, even their functional form is hidden from us. Without explicit equations, using ODE solvers for dynamical system modeling is out of reach.

Figure 1. In practice, it is common that the governing ODEs of many real-world dynamical systems are only partially known. (Image by author)

So how to address this issue?

One promising way is to adopt a data-driven mindset and leverage machine learning algorithms to infer the unknown dynamics from the observed data of the system states.

In this blog post, let’s explore one specific algorithm that can achieve this goal: Neural ODE. More specifically, we will first gain a conceptual understanding of how the Neural ODE approach can be used for dynamic modeling. Afterward, we will get hands-on and implement a Neural ODE model to address two case studies.

Disclaimer: As a tutorial, the Neural ODE implementation presented in this blog prioritizes simplicity and transparency. For more advanced implementations/libraries that are optimized for large-scale real-world challenges, please check out the reference I compiled at the end of this blog.

With that in mind, let’s get started!

Table of Content

· 1. What is Neural ODE?
· 2. Case Study Descriptions
∘ 2.1 Case study 1: System…



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