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Physics-informed neural networks for Timoshenko system with Thermoelasticity

Sabrine Chebbi, Joseph Muthui Wacira, Makram Hamouda, Bubacarr Bah

Abstract

The main focus of this paper is to analyze the behavior of a numerical solution of the Timoshenko system coupled with Thermoelasticity and incorporating second sound effects. In order to address this target, we employ the Physics-Informed Neural Networks (PINNs) framework to derive an approximate solution for the system. Our investigation delves into the extent to which this approximate solution can accurately capture the asymptotic behavior of the discrete energy, contingent upon the stability number $χ$. Interestingly, the PINNs overcome the major difficulties encountered while using the standard numerical methods.

Physics-informed neural networks for Timoshenko system with Thermoelasticity

Abstract

The main focus of this paper is to analyze the behavior of a numerical solution of the Timoshenko system coupled with Thermoelasticity and incorporating second sound effects. In order to address this target, we employ the Physics-Informed Neural Networks (PINNs) framework to derive an approximate solution for the system. Our investigation delves into the extent to which this approximate solution can accurately capture the asymptotic behavior of the discrete energy, contingent upon the stability number . Interestingly, the PINNs overcome the major difficulties encountered while using the standard numerical methods.
Paper Structure (12 sections, 2 theorems, 34 equations, 15 figures, 1 algorithm)

This paper contains 12 sections, 2 theorems, 34 equations, 15 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Physics Informed neural networks (PINNs) for Timoshenko system
  3. Numerical experiments
  4. Discussion about the neural network architecture
  5. PINNs for Timoshenko system with an exact solution
  6. PINNs on stability of Timoshenko system
  7. Energy decay rate analysis
  8. Non conservation of the energy, $\mu=0$.
  9. Linear damping $\mu \psi_t$.
  10. Non linear damping.
  11. Optimal logarithmic decay rate $\frac{\mu}{\psi_t} \cdot \exp\left(\frac{-1}{\psi_t^2}\right)$.
  12. Conclusion and future work

Key Result

Theorem 1

(2) For all initial data the system (1), cb and ci has a unique solution and

Figures (15)

  • Figure 1: Schematic diagram of the PINNs framework for Timoshenko systems.
  • Figure 2: The value of the total loss and PDE loss as a function of the number of epochs
  • Figure 5: Left:Thermoelastic Timoshenko beam $\mathbb{L}^2$ norm error in predictions; Right: Relative error
  • Figure 6: Left: the Total loss; right: the PDE, the boundary and the initial conditions' losses for case 1
  • Figure 7: Left: The Total loss; Right: The PDE boundary and the initial conditions' losses for case 2
  • ...and 10 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1