An efficient custom neural network framework for solving second order partial differential equations

Purpose This paper aims to develop an efficient neural network-based method for solving two-dimensional partial differential equations (PDEs) with prescribed initial and boundary conditions. The goal is to overcome the persistent challenge of balancing computational accuracy and efficiency faced by many existing neural network approaches, providing a robust and precise solution framework for engineering and scientific applications. Design/methodology/approach Following the classical constrained trial solution formulation, a custom three-layer network architecture is constructed in which the trial solution exactly satisfies all prescribed boundary and initial conditions through a parameter-free component. The PDE system is transformed into an unconstrained optimization problem governed solely by network parameters. An adaptive hidden neuron configuration strategy and a momentum-enhanced Levenberg–Marquardt training scheme are introduced to optimize network parameters by minimizing the system residual. Findings Numerical experiments on three distinct two-dimensional PDEs (elliptic with mixed boundary conditions, single boundary condition, and standard elliptic) demonstrate the method's superior accuracy. The proposed framework provides explicit closed-form approximate expressions for PDE solutions upon convergence, validating its robustness and precision. Research limitations/implications The current method is limited to two-dimensional, steady-state PDEs with regular domains and prescribed boundary conditions. Time-dependent problems, three-dimensional domains, and irregular geometries are not yet addressed. The adaptive hidden node strategy, while effective, requires exhaustive search over node counts, increasing computational overhead. Future research should extend the framework to transient phenomena, coupled multiphysics systems, and complex geometries, as noted in the conclusion. These extensions would broaden the applicability of neural network-based solvers in scientific computing. Practical implications The proposed method offers engineers a computationally efficient and accurate tool for solving PDEs without traditional mesh generation or numerical discretization. It provides explicit analytical solution expressions, facilitating sensitivity analysis, design optimization, and real-time simulations in fields such as structural mechanics, heat transfer, and fluid dynamics. The momentum-enhanced training ensures stable convergence, reducing trial-and-error tuning. This framework can be integrated into engineering software as a lightweight alternative to finite element or finite difference methods, particularly when rapid prototyping or repeated evaluations are required. Social implications Advancing PDE solution methods indirectly benefits society by enabling more efficient and reliable design of engineering systems, including transportation infrastructure, energy systems, and environmental modeling. Reduced computational costs lower energy consumption for large-scale simulations, contributing to sustainable computing practices. Improved accuracy in physical models can enhance safety and durability of civil structures. However, as a foundational numerical method, direct social impacts are mediated through specific applications. The open availability of the methodology (upon request) promotes broader access to advanced computational tools in research and education. Originality/value This paper introduces a novel neural network framework that transforms PDEs into an equivalent system governed exclusively by network parameters, avoiding traditional mesh-based discretization schemes. While the trial-solution structure follows the classical constrained formulation, the main innovations lie in the adaptive hidden neuron selection strategy and the momentum-enhanced Levenberg–Marquardt training scheme. Unlike existing methods that often sacrifice accuracy for efficiency, the proposed approach simultaneously achieves high precision and computational simplicity. The framework provides explicit analytical solution expressions upon convergence, offering a practical and highly accurate alternative to traditional PDE solvers for engineering applications.