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A-PINN: Auxiliary Physics-informed Neural Networks for Structural Vibration Analysis in Continuous Euler-Bernoulli Beam

Shivani Saini, Ramesh Kumar Vats, Arup Kumar Sahoo

TL;DR

The paper addresses solving high-order structural vibration problems governed by the Euler-Bernoulli beam equation using physics-informed neural networks. It introduces Auxiliary PINNs (A-PINN) with a balanced adaptive optimizer and auxiliary variables to decompose fourth-order PDEs into lower-order systems, improving stability and accuracy. Across three undamped vibration scenarios—free, forced, and Winkler-foundation vibrations—the A-PINN demonstrates superior fidelity to ground-truth solutions, faster convergence, and robustness to parameter perturbations, outperforming PINN, SANN, and FDM baselines. This approach offers a mesh-free, physically faithful framework for efficient, high-fidelity structural dynamics simulations with potential extensions to nonlinear, damped, and multi-physics settings.

Abstract

Recent advancements in physics-informed neural networks (PINNs) and their variants have garnered substantial focus from researchers due to their effectiveness in solving both forward and inverse problems governed by differential equations. In this research, a modified Auxiliary physics-informed neural network (A-PINN) framework with balanced adaptive optimizers is proposed for the analysis of structural vibration problems. In order to accurately represent structural systems, it is critical for capturing vibration phenomena and ensuring reliable predictive analysis. So, our investigations are crucial for gaining deeper insight into the robustness of scientific machine learning models for solving vibration problems. Further, to rigorously evaluate the performance of A-PINN, we conducted different numerical simulations to approximate the Euler-Bernoulli beam equations under the various scenarios. The numerical results substantiate the enhanced performance of our model in terms of both numerical stability and predictive accuracy. Our model shows improvement of at least 40% over the baselines.

A-PINN: Auxiliary Physics-informed Neural Networks for Structural Vibration Analysis in Continuous Euler-Bernoulli Beam

TL;DR

The paper addresses solving high-order structural vibration problems governed by the Euler-Bernoulli beam equation using physics-informed neural networks. It introduces Auxiliary PINNs (A-PINN) with a balanced adaptive optimizer and auxiliary variables to decompose fourth-order PDEs into lower-order systems, improving stability and accuracy. Across three undamped vibration scenarios—free, forced, and Winkler-foundation vibrations—the A-PINN demonstrates superior fidelity to ground-truth solutions, faster convergence, and robustness to parameter perturbations, outperforming PINN, SANN, and FDM baselines. This approach offers a mesh-free, physically faithful framework for efficient, high-fidelity structural dynamics simulations with potential extensions to nonlinear, damped, and multi-physics settings.

Abstract

Recent advancements in physics-informed neural networks (PINNs) and their variants have garnered substantial focus from researchers due to their effectiveness in solving both forward and inverse problems governed by differential equations. In this research, a modified Auxiliary physics-informed neural network (A-PINN) framework with balanced adaptive optimizers is proposed for the analysis of structural vibration problems. In order to accurately represent structural systems, it is critical for capturing vibration phenomena and ensuring reliable predictive analysis. So, our investigations are crucial for gaining deeper insight into the robustness of scientific machine learning models for solving vibration problems. Further, to rigorously evaluate the performance of A-PINN, we conducted different numerical simulations to approximate the Euler-Bernoulli beam equations under the various scenarios. The numerical results substantiate the enhanced performance of our model in terms of both numerical stability and predictive accuracy. Our model shows improvement of at least 40% over the baselines.
Paper Structure (14 sections, 34 equations, 16 figures, 15 tables, 1 algorithm)

This paper contains 14 sections, 34 equations, 16 figures, 15 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Statement
  4. The Description of A-PINN
  5. PINN Architecture
  6. Auxiliary Physics-informed Neural Networks
  7. Numerical Experiments
  8. Undamped Free Vibration of the Euler-Bernoulli Beam
  9. Undamped Force Vibration of the Euler-Bernoulli Beam
  10. Undamped Force Vibration of an Euler-Bernoulli Beam on a Winkler Foundation
  11. Discussion
  12. Conclusion
  13. Finite Difference Method
  14. Symplectic Artificial Neural Network

Figures (16)

  • Figure 1: Schematic illustration of the simply supported EBB for three undamped vibration cases: (a) free, (b) forced, and (c) forced with a Winkler foundation.
  • Figure 2: Architecture of physics-informed neural network
  • Figure 3: Architecture of Auxiliary physics-informed neural network
  • Figure 4: 2D comparison of physics-informed (top) and baseline (bottom) solutions for P1.
  • Figure 5: 3D comparison of physics-informed (top) and baseline (bottom) solutions for P1.
  • ...and 11 more figures