A pre-training deep learning method for simulating the large bending deformation of bilayer plates

TL;DR

This work tackles the challenge of simulating large bending deformations in bilayer plates by formulating a nonconvex energy minimization under an isometric constraint and solving it with a deep neural network. It introduces a penalty-based energy I[u]=E[u]+beta C[u]^2 and enforces boundary conditions exactly via a boundary-aware neural ansatz, while employing a novel pre-training strategy on nested subdomains to accelerate convergence toward the absolute minimizer. Across diverse geometries and curvatures, the method achieves high-accuracy results, maintains the isometric constraint with small error, and demonstrates the ability to reach absolute minimizers where gradient-flow methods may trap in local minima. These results indicate that deep learning offers a powerful, efficient alternative for challenging nonlinear elasticity problems with geometric constraints in bilayer plate systems.

Abstract

We propose a deep learning based method for simulating the large bending deformation of bilayer plates. Inspired by the greedy algorithm, we propose a pre-training method on a series of nested domains, which accelerate the convergence of training and find the absolute minimizer more effectively. The proposed method exhibits the capability to converge to an absolute minimizer, overcoming the limitation of gradient flow methods getting trapped in the local minimizer basins. We showcase better performance with fewer numbers of degrees of freedom for the relative energy errors and relative $L^2$-errors of the minimizer through numerical experiments. Furthermore, our method successfully maintains the $L^2$-norm of the isometric constraint, leading to an improvement of accuracy.

Figures (23)

• Figure 1: The component of ResNet.
• Figure 2: $\alpha = 1$ and $\beta = 500$. The pseudo-evolution of the bilayer plate reaches the absolute minimizer with energy $20$. The number of the iteration steps and the corresponding energy are reported.
• Figure 3: $\alpha = 2.5$ and $\beta = 1000$. We train the network directly and plot the evolution of the bilayer plate, and report the number of the iteration step and the corresponding energy. After about $3e5$ SGD iteration steps, the plate reaches a cylindrical shape.
• Figure 4: $\alpha = 5$, $\beta = 1000$. We train the network directly and plot the pseudo-evolution of the bilayer plate. The iteration steps and energy corresponding to the deformation are shown above. After about $1,200,000$ SGD iteration, the plate reaches a cylindrical shape, which is the absolute minimizer. The energy is $500$.
• Figure 5: Training loss, $\alpha= 5$. Decay of energy $E[\hat{u}]$ and $L^2$-isometric error $C[\hat{u}]$ for different penalized parameter $\beta$ during the training process.

Theorems & Definitions (13)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4
• Example 4.5
• Example 4.6