Physics-Informed Neural Networks for Nonlocal Beam Eigenvalue Problems

TL;DR

The paper tackles computing the first eigenpair of a sixth-order, stress-driven nonlocal beam model by embedding the governing equations and boundary conditions into a Physics-Informed Neural Network (PINN). By treating the eigenvalue as a trainable variable and enforcing constitutive and kinematic constraints along with a domain-wide normalization, the authors develop a robust unsupervised forward–inverse approach that yields eigenfunctions and eigenvalues in good agreement with analytical benchmarks. Key contributions include formulating a consistent stress-driven nonlocal beam theory, extending PINN methodology to sixth-order, incorporating constitutive boundary conditions, and providing detailed hyperparameter studies across local and nonlocal cantilever and simply supported beams. The approach offers mesh-free, scalable solutions with strong potential for rapid evaluation of complex nonlocal eigenproblems, albeit with higher computational cost due to high-order derivatives and sensitivity to training hyperparameters. Overall, the work demonstrates that PINNs can effectively address challenging high-order nonlocal eigenvalue problems with accurate predictions and practical implications for nano-structures and advanced beam theories.

Abstract

The present study investigates the dynamics of nonlocal beams by establishing a consistent stress-driven integral elastic using 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 boundary and constitutive boundary conditions. Relevant eigenvalues are treated as separate trainable variables. The results demonstrate that the proposed method is a powerful and robust tool for addressing the complexity of the problem. Once trained, the neural network is less computationally intensive than analytical methods. The obtained results are compared with benchmark analytical solutions and show strong agreement.

Figures (21)

• Figure 1: Schematic of Nonlocal Cantilever Beam
• Figure 2: PINN Architecture
• Figure 3: Local cantilever beam with high learning rate: $\lambda$ convergence, first eigenfunction, and loss histories.
• Figure 4: Influence of L2 regularization on PINN training - central part of the first eigenfunction curve. Colors: magenta - no L2 regularization, green - regularization value $10^{-4}$, blue - regularization value $10^{-6}$, red - analytical solution.
• Figure 5: Influence of gradient clipping on PINN training: central part of the first eigenfunction curve. Colors: green - no gradient clipping, magenta - no gradient clipping for $\lambda$ and NN weights gradients clipping value 1.0, cyan - clipping value 1.0, blue/dashed - clipping value 0.1, red - analytical solution.