# Optimizing Physics-Informed Neural Network in Dynamic System Simulation and Learning of Parameters
## 3 Artificial Neural Network
### Parameter Identification of Dynamical Systems Model Using PINN  
Let us consider a dynamical system (1). The goal is to use the PINN approach to solve and identify the parameters of the dynamical system model. This can be conducted by finding the best network parameters \(\omega\), representing the biases and weights network that minimizes the loss function. We offer a physics-informed neural network shown in Figure 1, with two networks to do this. The first network outputs are \(\theta(t_{n};\omega)\), which admits \(t\) as the input data. The second network learns the parameters \(\delta(t_{n};\omega)\) from the data. The following loss function was minimized. \[\xi(\omega;x)=\zeta_{t}(\omega;x_{t})+\zeta_{r}(\omega;x_{r}) \tag{2}\]\(\zeta_{t}(\omega;\kappa_{t})\) is called the training loss and \(\zeta_{r}(\omega;\kappa_{r})\) is called the residual loss. \(\kappa=\kappa_{r}\cup\kappa_{t}\) in the total training datasets. We define the following:  
\[\zeta_{r}(\omega;\kappa_{r})=\frac{1}{\kappa_{r}}\sum_{i=1}^{p}\sum_{n=1}^{ \kappa_{r}}|r_{i}(t_{n})|^{2} \tag{3}\]  
where  
\[r_{i}(t_{n})=\frac{d\theta(t_{n};\omega)}{dt}-\Theta\bigg{(}t_{n},\theta(t_{n };\omega),\delta(t_{n};\omega)\bigg{)} \tag{4}\]  
and  
\[\zeta_{t}(\omega;\kappa_{t})=\frac{1}{\kappa_{t}}\bigg{(}\sum_{n=1}^{\kappa_ {t}}||\theta(t_{n};\omega)-\theta(t_{n})||^{2}\bigg{)} \tag{5}\]  
The network is trained to minimize (3); an optimizer such as Adams is the most optimal weight and bias that would minimize (3). The neural network also obtains the optimal values for the model parameters. The \(\theta(t_{n})\) represents the loss function's training data. We identify the parameter \(\delta\) by solving Equation (5). When trying to address the problem (5), we can categorize it into two possible scenarios:  
Scenario 1: In this particular instance, we assume the right-hand side function of a dynamical system (1) has an equal number of parameters and equations. This scenario shows that a solution exists. In other words, there is an exact solution [12]. Scenario 2: In this context, the number of parameters does not match the number of dynamical system equations [12]. **Remark 1**. : _It is important to underscore that the number of networks can exceed two, contingent upon the intricacies of the dynamical system and the parameters involved._  
The implementation of physics-informed neural networks (PINNs) is carried out using Python. Its simplicity and extensive library ecosystem make it an ideal choice for developing sophisticated models like PINNs. Key to our implementation is TensorFlow, which offers robust tools for building and training neural network models. TensorFlow's compatibility with Python and its efficient handling of large-scale numerical computations enable the effective construction of PINN architectures. Additionally, we employ NumPy, a Python library that supports large, multi-dimensional arrays and matrices, along with a collection of high-level mathematical functions to operate on these arrays. This is crucial  
Figure 1: Schematic diagram of the PINN with the parameters of a dynamical system of ODE model. for handling the numerical aspects inherent in PINN development. We also utilize Pandas, a library for data manipulation and analysis. Pandas provides structures and functions that are ideal for modifying numerical tables and time series, making it useful for preprocessing and organizing the data that neural networks need to learn. The synergy of these libraries in Python creates a powerful toolkit for developing and implementing PINNs. The source code in Figure 2 provides a comprehensive example of physics-informed neural network implementation, illustrating the practical application of parameter identification in dynamical systems. This Python-based code features a custom PINN class, which encompasses the complete workflow of constructing, training, and deploying the neural network for predictive analysis.