# Optimizing Physics-Informed Neural Network in Dynamic System Simulation and Learning of Parameters
## 5 Computational Simulations of Dynamical Systems
### Brusselator Model  
The Brusselator model is a simple mathematical model for the behavior of chemical oscillations. [PERSON] and [PERSON] first proposed it in 1968 [26]. The model comprises two chemical species, U and V, that interact through non-linear differential equations. The equations describe how the concentrations of the two species change over time, taking into account chemical reactions, diffusion, and other factors. The Brusselator is known for its ability to make complex, self-sustaining oscillations in the concentrations of the two chemical species. As a result, it has been used to study a wide range of phenomena in chemistry, biology, and physics, including patterns of gene expression and the behavior of simple electronic circuits [27; 28]. The Brusselator model is useful in various fields because it is a simple yet rich model that can exhibit a wide range of complex behaviors. The Brusselator reaction is a simple model of autocatalytic chemical reactions, which is a chemical reaction catalyzed by one of the products of the reaction. The reaction has been used as a model to study the emergence of self-organizing behavior in chemical systems [29]. It has been applied to various fields, including chemistry, physics, biology, and engineering. The Brusselator reaction consists of the following:  
\[A\longrightarrow U\rightarrow\frac{du}{dt}=k_{a}a\]  
\[2U+V\longrightarrow 3U\rightarrow\left(\frac{du}{dt}=k_{b}u^{2}v,\frac{dv}{dt}=-k _{b}v^{2}y\right)\]  
\[B+U\longrightarrow V+C\rightarrow\left(\frac{du}{dt}=-k_{c}bu,\frac{dv}{dt}=- k_{c}bu\right)\]  
\[U\longrightarrow D\rightarrow\frac{du}{dt}=-k_{a}u.\]  
The overall reaction is \(A+B\longrightarrow C+D\) with intermediary species \(U\) and \(V\). \(A\) and \(B\) are reactants while \(C\) and \(D\) are products. During the chemical reaction, suppose the concentration reactant \(A\) and \(B\) are kept constant; therefore, the ordinary differential equation version of the Brusselator is given below:  
\[ \begin{split}\frac{du(t)}{dt}&=a(t)+u(t)^{2}v(t)-(b(t )+1)u(t)\\ \frac{dv(t)}{dt}&=bu(t)-u(t)^{2}v(t).\end{split}  \tag{12}\]  
where \(u(t)\) and \(v(t)\) are the concentrations of the two chemical species involved in the reaction, and \(a(t)\) and \(b(t)\) are the time-varying parameters that control the system's behavior. The Brusselator model exhibits various dynamic behaviors, depending on the values of the \(a\) and \(b\) parameters [26]. They demonstrated that the system could exhibit stable, steady states, oscillations, and chaotic behavior. Finally, the parameters of the Brusselator system are also important in understanding the behavior of real-world chemical systems, which can be modeled using similar equations. Understanding how the parameters of such a system affect its behavior can provide insights into the underlying chemical reactions and help predict and control the system's behavior. Our primary aim is to learn the time-varying parameters of the Brusselator model. Here, we used synthetic data that was generated by randomizing the numerical solution of the Brusselator model derived from an ODE-solver on the Brusselator model, where we assumed the initial conditions for the first species and the second species variables to be \(u(0)=1\) and \(v(0)=1\), respectively. We choose \(a=1.0\) and \(b=2.5\). This problem was solved on the time interval \([0.T]\), where T = 30. We used a grid with \(P_{x}=1000\) points on the time interval, and every second-time step, the numerical solution vector is taken to obtain the observation or synthetic data. We use the PINN architecture shown in Figure 1 with three networks to learn the time-varying parameters of the Brusselator model. The first network outputs are \(u_{NN}(t_{n};\omega)\) and \(v_{NN}(t_{n};\omega)\), which admits \(t\) as the input data. The second and third networks learn the parameters \(a(t_{n};\Phi)\) and \(b(t_{n};\Phi)\) from the data.