A Mesh-Free Physics-Informed Neural Network Based on Rayleigh Quotient for Structural Frequency Analysis of Beams and Plates

Accurate estimation of natural frequencies is essential in structural engineering applications involving beams and plates. Traditional finite element methods are computationally expensive, especially for complex geometries. This paper presents an unsupervised, mesh-free framework based on Physics-Informed Neural Networks (PINNs) that leverages the Rayleigh quotient as a variational loss functional. The proposed method models the mode shape using neural networks while minimizing the ratio of total strain energy to kinetic energy. The approach accurately predicts the fundamental frequencies of Euler-Bernoulli beams and Kirchhoff--Love plates under various classical boundary conditions. Benchmark comparisons with analytical solutions show relative errors as low as $10^{-5}$. The method converges rapidly, requiring fewer collocation points and less training time. This is the first unsupervised PINN framework based on Rayleigh quotient to achieve near-zero error in fundamental frequency extraction for both 1D and 2D structural domains. The method generalizes across different boundary conditions and geometries and offers a scalable alternative to traditional solvers.