Causal Physics-Informed Neural Networks for Singular and Singularly Perturbed Boundary Value Problems in Chemical Systems

Singular boundary value problems (SBVPs) and singularly perturbed boundary value problems (SPBVPs) are commonly arise in the modeling of critical chemical processes, including isothermal gas spheres, electroactive polymer films, thermal explosions, and chemical reactor theory. Due to their extensive applications, these problems require the creation of effective numerical methods for precise solutions. Traditional methods often struggle with stiffness and multiscale behavior, while analytical solutions are limited to specific cases. In this work, we use a data-driven approach using Physics-Informed Neural Networks (PINNs) to solve SBVP and SPBVPs. By initializing the perturbation parameter as a trainable variable, it is optimized alongside net-work weights and biases during training. Numerical simulations demonstrate that causality-enhanced PINNs significantly improve the solution accuracy for singular chemical boundary value problems.