Stoichiometric modeling, existence, stability, and optimal control of multi-species sodium carbonate formation using cPINNs with PDFP optimization

In this study we present a comprehensive mathematical, analytical, and computational investigation of the sodium carbonate formation process that are governed by a strongly coupled nonlinear chemical kinetics system of ODEs. A detailed reaction mechanism that involves twelve interacting chemical species are formulated as a system of ordinary differential equations that is analytically tractable and suitable for modern learning based solvers, specially physics informed neural network. Fundamental qualitative properties of the model are rigorously examined, including existence and uniqueness of solutions, boundedness, positivity, chemical reaction free equilibrium points, reaction generation numbers, Lyapunov stability of steady states, and sensitivity with respect to kinetic parameters. To regulate the buildup of reactive intermediates, an optimal control framework is introduced by incorporating two bounded time dependent control functions into selected reaction pathways, and the existence of an optimal control solution is established theoretically. The resulting uncontrolled and controlled models are solved using a Physics Informed Neural Network framework, which constructs smooth global approximations of all state variables without relying on conventional time marching schemes. All kinetic rate constants are treated as learnable parameters, enabling simultaneous state prediction and parameter identification through automatic differentiation. Training efficiency and numerical robustness are enhanced through the use of a Preconditioned Davidon–Fletcher–Powell quasi Newton optimizer, which is particularly effective for stiff and highly nonlinear reaction dynamics. We present the numerical experiments using chemically consistent initial conditions that are derived from stoichiometric relations, as well as randomly generated initial states to examine the generalization. The dynamical behavior of the reaction network is also analyzed using concentration profiles, time derivative evolution, phase plane, and configuration space representations. In the optimal control case, the learned control strategies are studied in detail, that show the adaptive regulation of reaction pathways that maintains the stability and efficiency. The results confirmed the reliability, accuracy, and physical consistency of the cPINNs based framework for complex chemical kinetics and optimal control problems.