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.

The analytical results presented here not only deepen the understanding of coupled magneto-bioconvective transport phenomena but also highlight the possibility of various applications including microelectronic cooling, renewable energy systems, electromagnetic flow control, biomedical transport, microbial fuel cells, and advanced nanofluid-based thermal technologies. The present study investigates a three-dimensional Casson nanofluid flow over a Riga surface at stagnation point under the influence of an applied magnetic field, an exponential heat source, and a rotating frame. This study explores how these combined physical mechanisms influence velocity, temperature, nanoparticle concentration, and microorganism distributions. Also, it assesses whether the Homotopy analysis method (HAM) is capable of yielding precise analytical solutions for such a highly nonlinear transport model. The original nonlinear partial differential equations representing magneto-bioconvective Casson nanofluid flow are first converted to a dimensionless system of ordinary differential equations by using appropriate similarity transformations. The coupled system thus obtained is then solved analytically by the HAM. The solutions achieved through this method are checked against results from the literature to ensure their validity. The finding shows that enhancement in the Casson fluid parameter, magnetic parameter, and mass Grashof number leads to a notable decrease in velocity field as a result of increased flow resistance. In contrast, the higher Hartmann numbers produced by the Riga surface aid fluid motion via electromagnetic forcing. A stronger heat source and larger Biot number cause temperature distribution to rise, whereas thermophoresis lowers nanoparticle concentration. Also, higher activation energy affects concentration transport, but an increase in Peclet number boosts microorganism distribution and bioconvection strength.