Physics-informed neural networks represent a category of deep learning models that directly incorporate physical laws into the training process to solve differential equations, thereby diminishing the dependence on extensively labeled datasets. This study investigates a constrained optimization framework within PINNs to address singular three-point boundary value problems, which present significant challenges owing to singularities and internal boundary conditions that result in non-standard solution behavior. To address these complexities, we developed a customized Physics-informed neural network architecture that integrates constraint-driven regularization terms into the loss function to enhance the generalization and numerical stability. The proposed approach was evaluated across multiple benchmark problems, with performance assessed using statistical metrics and the mean squared error. The optimization and training PINN regular framework will stabilize the training and convergence in the presence of singularities to yield dependable TPS-BVP solutions. The predicted solutions were rigorously compared with exact analytical solutions. The results demonstrate that the constrained optimization-based Physics-informed neural networks framework provides highly accurate and stable approximations, validating its effectiveness in handling complex singular boundary value problems.
Nadeem et al. (Mon,) studied this question.