## 4 Digital Twins Derivation
### Structure of Digital Twins  
This paper utilizes and compares four types of different methods to construct the digital twins of the cooling fan system and evaluate each modeling performance to find the best. Four types of models are the physical model, the NARX model, the neural ODE model, and the hybrid neural ODE model. The models' structure and number of parameters can be seen in Table 1. All the neural network models only include a single hidden layer. Although increasing the number of hidden layers may improve fitting results, it also presents challenges in real-time implementation with limited memory storage. In addition, the paper only sets applied torque and previous fan speed as the inputs of the digital twins. Since the proposed ODE model is designed to describe the dynamics of the cooling fan, the inputs of the model also have to follow the physical laws governing the identified system. Although increasing more input features may enhance the fitting performance, it runs the risk of reducing the model's ability to generalize the fan dynamics effectively. Furthermore, according to the practical scenario, fan speed and applied torque are the only available measurements from the experimental device. These are the main reasons why this research only selected these key parameters that have physical meanings as the inputs of the model. The directly measurable information not only reduces the model's complexity but also ensures the interpretability of the digital twin modeling. In this study, two sets of data are used for training and testing: training data are used to estimate the parameters of the digital twins, and testing data are utilized to examine the fitting result.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Model** & **Parameter Estimation** & **Structure** & **No. of** \\
**Type** & **Method** & **Parameters** & **Details** \\ \hline Physical & Filtering integral & \(\dot{\omega}(t)=\frac{1}{J}[-a\omega(t)-\mathrm{C_{D}}\omega^{2}(t)+\tau(t)]\) & 3 & – \\ model & operator & \(\omega(t)=\omega(0)+\int_{0}^{t}\dot{\omega}(t)dt^{\prime}\) & \\ [PERSON] & Levenberg– & \(\omega(t)=\textbf{B}_{o}+\textbf{W}_{o}\cdot\mathrm{t}\mathrm{a}\mathrm{i} \delta\left(\begin{array}{c}\textbf{B}_{i}+\textbf{W}_{i}\textbf{X}_{D}(k) \\ +\textbf{W}_{i}\textbf{Y}_{D}(k)\end{array}\right)\) & 2051 & 10 th order with 50 hidden states \\ Neural & ADAM optimization & \(\dot{\omega}(t)=f(\omega(t),\tau(t),\theta)=\textbf{W}_{o}\mathrm{t}\mathrm{a} \mathrm{i}\delta\left(\begin{array}{cc}\omega(t)&\tau(t)\end{array}^{T}+ \textbf{b}_{i}\right)+\textbf{b}_{o}\) & 81 & 1 st order with 20 hidden states \\ Hybrid & \(\omega(t)=\omega(0)+\int_{0}^{t}f(\omega(t^{\prime}),\tau(t^{\prime}),\theta) dt^{\prime}\) & & 1 st order with 10 hidden states for \\ neural & ADAM optimization & \(\dot{\omega}(t)=\frac{1}{J}[-a\omega(t)-\mathrm{C_{D}}\omega^{2}(t)+\tau(t)]+f (\omega(t),\tau(t),\theta)\) & 34 & 1 st order with 10 hidden states for \\ ODE & & \(\omega(t)=\omega(0)+\int_{0}^{t}\dot{\omega}(t^{\prime})dt^{\prime}\) & & NN part \\ \hline \end{tabular}
\end{table}
Table 1: The details of four types of cooling fan digital twins.  
Figure 8: Physically based model fitting performance in high-speed conditions.