A Mixed Formulation Physics-Informed Neural Network for a Forward and an Inverse Euler–Bernoulli Dynamic Beam Problem

A mixed formulation physics-informed neural network (PINN) for solving a forward and an inverse nonlinear Euler–Bernoulli dynamic beam problem is proposed. This approach avoids the computation of fourth-order spatial derivatives in the loss function, thereby mitigating training instability and allowing seamless integration of dynamic forward and inverse problems. A sensitivity analysis is provided for the investigation of the uniqueness of the solution in the inverse problems. The developed programming code is based on open-source Python software 3.13 for deep learning. The impact of the structural hyperparameters of the PINN on the results is shown. Several numerical examples are presented. The error of approximation and the results of the convergence are given, alongside a comparison of the neural numerical solutions with the exact solutions. The inverse problem involving noisy and noise-free data has also been tested.