# Optimizing Physics-Informed Neural Network in Dynamic System Simulation and Learning of Parameters
## 1 Introduction  
In many scientific and engineering fields, understanding and predicting a wide range of complex events is all about understanding and predicting dynamical systems. These systems often employ ordinary differential equations (ODEs) to describe and model the intricate relationships inherent in various natural and artificial processes. One such common usage is seen in compartmental models, extensively used across fields such as ecology [1], epidemiology [2], chemical engineering [3], mathematical biology [4], and economics. These models require careful estimation of numerous unknown parameters for their validity, often derived from data collected from experiments or real-world observations. Thisprocess, termed calibration of the dynamical system, constitutes a challenging optimization problem due to its iterative nature and potential convergence issues. It is also important to keep the gap between observed data and model forecasts as small as possible. This makes the accurate estimation of parameters a key step in developing models. Moreover, the need for dynamical system models with accurate parameter estimates in chemical process engineering has surged due to process optimization and control technology advancements [5]. However, the difficulties associated with parameter estimation become compounded in models, such as ODEs or differential-algebraic equations (DAEs), that exhibit non-linearity in parameters. While many system identification techniques such as the Gauss-Newton method [6], multiple shooting, recursive estimation [7], maximum likelihood estimation, Markov chain Monte Carlo-based Bayesian inference [8; 9], finite element methods [10], and collocation methods [11] are available for learning parameters and solving dynamical systems, each has limitations, including high computational costs, reliance on initial guesses, and limitations with time-varying parameters. Despite the challenges, the role of parameter estimation in model accuracy and predictability remains paramount, calling for continued innovation and research in this fundamental mathematical field. Artificial intelligence (AI), particularly the implementation of deep neural networks (DNNs), has experienced a significant increase in applications for analyzing and interpreting complex systems [12; 13]. This is especially true for dynamical systems, which present complex interactions requiring sophisticated analytic tools. Although these AI models have demonstrated remarkable efficacy in data fitting and short-term prediction tasks, their implementation may need to be improved due to certain obstacles. One major barrier is that it takes work for AI models to understand the underlying structures of a dynamic system's activities. The resulting AI model predictions and real system behaviors may be at odds with one another. Additionally, the performance of these models is heavily reliant on the quantity and quality of input data, potentially resulting in inadequate insights when the data fails to portray the system's dynamics. Artificial neural networks (ANNs) have gained popularity as an AI model because of their potential to overcome common problems in the field, such as the necessity of massive training datasets and high processing costs. It has been proposed that ANNs can be used to pioneering effect in estimating parameters in systems of differential equations. The advent of ANNs traces back to the 1940s, primarily attributed to the research article by [PERSON] and [PERSON] [14]. However, it is only in recent years that ANNs and DNNs have seen widespread acceptance and use, owing to significant advancements in computational capabilities and data storage. DNNs, essentially ANNs with multiple hidden layers, have seen a surge in application across different sectors, including computer vision, image processing [15], pattern recognition [16], and cybersecurity [17]. The architecture of DNNs allows them to capture more variance, contributing to their success.