Physics-Informed Neural Networks Versus Differential Transform Method for Reduced Second-Order ODEs in Membrane Shell Theory

This paper presents a comparative study of two approaches for solving second-order ordinary differential equations arising from the membrane theory of shells of revolution. The considered equations originate from the rotational symmetry of shell structures, which enables the reduction of the governing partial differential equations to a sequence of ordinary differential equations corresponding to individual circumferential harmonics. The study compares the classical Differential Transform Method (DTM) with Physics-Informed Neural Networks (PINNs). Both initial value and boundary value problems are investigated, including benchmark examples with known analytical solutions and a systematic analysis of the influence of PINN architecture on the solution accuracy. For the PINN approach, the effects of the number of collocation points, hidden layers, and neurons per layer on the approximation error and training time are examined. The results demonstrate that DTM provides an efficient framework for constructing analytical solutions of initial value problems with minimal computational cost. However, its application to boundary value problems requires the introduction of additional auxiliary parameters and the solution of supplementary nonlinear equations, considerably increasing the analytical complexity of the procedure. In contrast, PINNs achieve high accuracy for both initial and boundary value problems while naturally incorporating boundary conditions through the loss function. The presented results demonstrate how the exploitation of geometric symmetry, combined with modern scientific machine learning techniques, provides an effective computational framework for solving differential equations arising in shell mechanics.