## 1 Introduction  
Cooling fan systems have been widely used in various industrial fields, such as nuclear power plants [1], automobiles [2], and workstation computers [3]. They are often used to maintain temperature and ensure an optimal operation environment. Due to their low development cost, efficient heat dissipation, and air circulation, a cooling fan system is often regarded as the primary solution for environmental temperature control use [4, 5]. Despite playing an important role in the manufacturing process, the rotary mechanism of the cooling fan is highly susceptible to failure, making it one of the top 10 failing components in electronic products [6]. Therefore, it is crucial to monitor the condition of the cooling fan system to prevent drastic changes in the operation environment. To achieve accurate condition monitoring during the manufacturing process, it is necessary to create the digital twins of a system, which provide the virtual representation of the identified physical entity. According to the data from sensors, digital twins can update their state to mirror the dynamic behavior and the condition of the system in real time. By incorporating other analysis tools, the digital twin model can be used not only for condition monitoring, but also for prognosis, optimization, and control in the digital world [7]. Therefore, digital twins are also regarded as the backbone of Industry 4.0 [8, 9, 10, 11], providing the necessary and useful information to enhance productivity and making intelligent manufacturing smarter, more efficient, and more convenient [12]. Digital twins can be created using various kinds of methods, including using physical models or data-driven approaches. Physical methods require domain knowledge of the identified system. For an industrial cooling fan, aerodynamics, blade geometry, and the control theorem need to be considered. There have been several works analyzing its dynamics from previous studies [4; 5; 13]. Since the digital twin is modeled using physical principles, it has better robustness and interpretation of the anomalies. However, the result may still have a large deviation between the identified systems because the physical equation is usually built based on ideal conditions. Therefore, the digital twin may not be able to reflect the dynamic behavior in the real world. In contrast, data-driven methods directly employ feature extraction to find the dynamical pattern from the historical data. Common methods include observer/Kalman filter identification [14; 15], dynamic mode decomposition (DMD) [16; 17], and sparse identification of nonlinear dynamics (SINDy) [18; 19]. The most well-known and commonly used method is the neural network (NN), which provides a structure that can approximate any system dynamics based on the universal approximation theorem [20]. As the NN has evolved, different layers and structures have been developed. One of the NN types is called a recurrent neural network (RNN), which is renowned for modeling sequential data. Since system dynamics signals can also be seen as time-series data, RNN has been proven to perform well in system modeling [21; 22; 23]. However, applying RNN or any other data-driven method requires a large amount of fruitful inputs/outputs to avoid unstable model predictions [24; 25]. Furthermore, most of the system dynamics are better described in the continuous-time domain. These reasons lead to excessively large model sizes and memory storage usage when employing RNN for robust modeling, which may pose a challenge in applying digital twins for real-time use. To explore an alternative way to create the digital twins, this paper proposes using neural ordinary differential equations (neural ODEs) to approximate the governing equations of the system. The concept of neural ODEs establishes a viewpoint to connect the deep learning model with differential equations [26], offering a more flexible description and reducing the need for numerous parameters for modeling system dynamics.