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<p>The present study assesses the performance of physics-informed neural networks (PINNs) for beam bending problems under systematic variation of key hyperparameters, including activation functions, learning-rate schedulers, and collocation strategies. One-dimensional beam models serve as benchmarks, covering linear Euler–Bernoulli and Timoshenko beams with analytical solutions, as well as nonlinear, functionally graded, and hyperelastic cases validated against the differential quadrature method (DQM) and finite element method (FEM). The obtained outcomes show that the Swish and Tanh activation functions provide accurate predictions for displacement and derivative quantities, with Swish achieving lower loss values. In contrast, ReLU fails to converge for the considered problems, and Adaptive Tanh shows strong sensitivity to its trainable scaling parameter, leading to unstable higher-order derivative predictions in fourth-order beam formulations. Furthermore, good agreement is observed between the PINN predictions and all the reference solutions, with low errors for both linear and nonlinear models. The present investigation provides a systematic assessment of PINN behavior for beam bending problems governed by higher-order PDEs, with particular attention to the reliability of derivative predictions essential for structural quantities such as bending moments and shear forces. Unlike most existing PINN beam studies that focus primarily on displacement prediction, the present assessment systematically validates slope, bending moment, shear force, and contact pressure fields in addition to displacement. Repeated training under different random initializations shows low variability in the predicted displacement field despite considerable variation in residual loss magnitude, indicating limited sensitivity of the physical solution to stochastic weight initialization. While classical numerical methods remain more computationally efficient for one-dimensional forward analyses, the present results indicate that PINNs provide an accurate mesh-free approximation framework for the investigated beam problems, which may support future investigations of inverse, parameter identification, and coupled multi-physics structural problems.</p>

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Assessment of physics-informed neural networks for beam bending problems in structural mechanics

  • 期刊论文

  • 发表日期：

    2026年7月1日

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  • Ali Forooghi,
  • Sajjad Miralizadeh Jalalat,
  • Tito Andriollo,
  • Eugenio Ruocco

摘要

The present study assesses the performance of physics-informed neural networks (PINNs) for beam bending problems under systematic variation of key hyperparameters, including activation functions, learning-rate schedulers, and collocation strategies. One-dimensional beam models serve as benchmarks, covering linear Euler–Bernoulli and Timoshenko beams with analytical solutions, as well as nonlinear, functionally graded, and hyperelastic cases validated against the differential quadrature method (DQM) and finite element method (FEM). The obtained outcomes show that the Swish and Tanh activation functions provide accurate predictions for displacement and derivative quantities, with Swish achieving lower loss values. In contrast, ReLU fails to converge for the considered problems, and Adaptive Tanh shows strong sensitivity to its trainable scaling parameter, leading to unstable higher-order derivative predictions in fourth-order beam formulations. Furthermore, good agreement is observed between the PINN predictions and all the reference solutions, with low errors for both linear and nonlinear models. The present investigation provides a systematic assessment of PINN behavior for beam bending problems governed by higher-order PDEs, with particular attention to the reliability of derivative predictions essential for structural quantities such as bending moments and shear forces. Unlike most existing PINN beam studies that focus primarily on displacement prediction, the present assessment systematically validates slope, bending moment, shear force, and contact pressure fields in addition to displacement. Repeated training under different random initializations shows low variability in the predicted displacement field despite considerable variation in residual loss magnitude, indicating limited sensitivity of the physical solution to stochastic weight initialization. While classical numerical methods remain more computationally efficient for one-dimensional forward analyses, the present results indicate that PINNs provide an accurate mesh-free approximation framework for the investigated beam problems, which may support future investigations of inverse, parameter identification, and coupled multi-physics structural problems.