The Lane-Emden equations are essential tools for modeling heat and mass transfer, chemical reactions, and various scientific fields. This study focuses on solutions to Lane-Emden equations in chemical engineering. It examines the impact on concentration profiles in both catalyst and biocatalyst systems with cylindrical and spherical geometries. The research proposes a hybrid approach combining a collocation method with genetic algorithms to solve Lane-Emden equations in diffusion-reaction systems. It also investigates how different parameters, such as Thiele modulus (ρ), dimensionless activation energy (µ), and dimensionless heat of reaction (α) affect these solutions. The methodology is effective across both low and high values of ρ, µ, and α. Overall, the results demonstrate the potential to address limitations of previous methods and highlight the strength of the GA-based approach.

Differential equations govern the dynamics of many physical systems, including mass–spring and mass–spring–damper systems. We propose a fully connected Gudermannian-activated neural network (FCGNN) trained with a hybrid global–local optimizer: a genetic algorithm for a global search, followed by an active-set method for rapid local refinement. The Gudermannian activation smoothly links trigonometric and hyperbolic behaviors, enabling a single network to capture both oscillatory (underdamped/forced) and exponential (overdamped) responses with improved expressivity over standard activation functions. We also studied the effect of L1 and L2 regularization on generalization. Using canonical vibration benchmarks, Monte Carlo trials quantify the robustness of the initialization and noise. The FCGNN’s predictions closely match the analytical solution and outperform physics-informed neural networks, achieving higher accuracy with compact architectures and reduced training effort. The novelty of the model is in the integration of the gudernmannian activation function with a hybrid GA–ASM optimization and independent regularization analyses, along with the Monte Carlo simulations, which yield an accurate solver for oscillatory systems.

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.