Physics-informed neural networks for nonlocal beam eigenvalue problems

The present study investigates the dynamics of stress-driven nonlocal elastic beams exploiting the Physics-Informed Neural Network (PINN) approach. Specifically, a PINN is developed to compute the first eigenfunction and eigenvalue arising from the underlying sixth-order ordinary differential equation. The PINN is based on a feedforward neural network, with a loss function composed of terms from the differential equation, the normalization condition, and both classical boundary and constitutive boundary conditions. Relevant eigenvalues are treated as separate trainable variables. The results demonstrate that the proposed method is a powerful tool for addressing the complexity of the problem. The obtained results are compared with benchmark analytical solutions and show strong agreement.