Physics-informed neural networks framework for solving the highly nonlinear Bratu equation arising in combustion theory

This research introduced a physics informed neural network (PINN) framework designed to effectively solve the highly nonlinear Bratu equation, which arises in various physical contexts such as chemical reaction theory and combustion processes. PINNs provide a mesh-free solution by embedding physical laws directly into the loss function of the neural network and utilizing automatic differentiation for accurate derivative calculations. However, standard PINNs often face challenges in strictly enforcing boundary conditions (BCs), resulting in numerical inaccuracies and slow convergence. To overcome this, we proposed an innovative method that precisely enforces BCs through a transformation, thereby eliminating residual errors and significantly improving the reliability and performance of the PINN framework. Numerical experiments validated the effectiveness of the proposed approach, showing improved accuracy, faster convergence, and more stable training dynamics. Detailed analyses were conducted to investigate the influence of key hyperparameters, such as activation functions, network architecture, and learning rates, on the model's performance.