# Optimizing Physics-Informed Neural Network in Dynamic System Simulation and Learning of Parameters
## 1 Introduction
The effectiveness of compartmental models, which are often described by systems of ordinary differential equations, and the effectiveness of AI models in understanding and predicting the behaviors of dynamical systems have been demonstrated. However, they are not without their weaknesses. Researchers are increasingly gravitating toward combining compartmental and AI models to improve their abilities to analyze and predict the behavior of dynamical systems. The recent emergence of physics-informed neural networks (PINNs) has demonstrated a promising stride in integrating differential equations into neural networks [18]. This allows the networks to fit the data accurately, learn the parameters, and adhere to the equations. This approach employs neural networks to simulate nonlinear systems while minimizing the necessary data and limiting the model's search space with pre-existing knowledge, such as a system of differential equations. Since this development, physics-informed neural networks (PINNs) have been routinely employed as a method for nonlinear function approximation. These networks have exhibited impressive capabilities in tackling diverse scientific computing tasks across multiple fields. Moreover, various research initiatives have utilized the PINNs framework for modeling and scrutinizing dynamic systems. [PERSON] et al. introduced the logistic-informed neural network (LINN), inspired by the physics-informed neural network (PINN), to specifically predict the number of individuals infected by the COVID-19 Omicron variant. The major benefit of using LINN in their study lies in its ability to incorporate logistic growth behavior directly into the neural network, resulting in predictions that align well with the natural growth pattern of an epidemic. This method provides more accurate and reliable forecasts for the Omicron variant spread, crucial for timely and effective public health responses and resource allocation. By integrating logistic growth principles, LINN ensures that the model's predictions are not only data-driven but also grounded in epidemiological knowledge [19]. The PINN algorithm was used to learn the epidemiological parameters of a model for COVID-19 vaccine efficacy [20]. [PERSON] et al. applied PINN to identify and predict the time-varying parameter of the susceptible (S), infected (I), recovered (R), and death (D) (SIRD model) [21], [PERSON] et al. used the epidemiological-informed neural network (EINN), which was developed through PINN, to learn the time-varying transmission rate for the COVID-19 pandemic in various mitigation scenarios [22]. Thus, this paper seeks to make the following significant contributions:  
1. We employ physics-informed neural networks (PINNs) to learn the unknown time-varying parameters within systems of differential equations. 2. Although the application of PINNs in modeling dynamic systems is well-established, our work introduces novel approaches that substantially augment the existing methodology. Firstly, we present a unique assumption where the right-hand side function of a dynamical system is posited to have an equal number of parameters and equations. This hypothesis is pivotal because it implies the existence of an exact solution for the system, a scenario rarely considered in the existing literature. By demonstrating this principle, we provide a foundational understanding that can be leveraged for more accurate modeling of certain types of dynamical systems. Secondly, we address the more complex situation where the number of parameters does not align with the number of equations in the dynamical system. Here, we focus on developing and optimizing network architectures that significantly enhance the convergence speed. This innovation results in improved computational efficiency and elevates the accuracy in estimating time-varying parameters. Our approach is particularly effective in complex systems where traditional models struggle with computational load and accuracy. 3.