# A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches

arXiv: https://arxiv.org/abs/2010.06050 · HTML: https://mlsanity.com/paper/2010.06050/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Oct 2020.

## Built on by (most cited first, 30 of 34)

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- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
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- [Identifying Constitutive Parameters for Complex Hyperelastic Materials using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.15640/) (2023)
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- [A Hybrid Probabilistic Battery Health Management Approach for Robust Inspection Drone Operations](https://mlsanity.com/paper/2405.00055/) (2024)
- [Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks](https://mlsanity.com/paper/2405.18311/) (2024)
- [Utilising physics-guided deep learning to overcome data scarcity](https://mlsanity.com/paper/2211.15664/) (2022)
- [Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization](https://mlsanity.com/paper/2412.04596/) (2024)
- [PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs](https://mlsanity.com/paper/2607.20378/) (2026)
- [Semi-supervised physics guided deep learning framework for predicting the I-V characteristics of GAN HEMT](https://mlsanity.com/paper/2110.10724/) (2021)
- [Interpretability and accessibility of machine learning in selected food processing, agriculture and health applications](https://mlsanity.com/paper/2211.16699/) (2022)
- [Physics-informed radial basis network (PIRBN): A local approximating neural network for solving nonlinear PDEs](https://mlsanity.com/paper/2304.06234/) (2023)
- [Physics-informed neural networks for blood flow inverse problems](https://mlsanity.com/paper/2308.00927/) (2023)
- [Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials](https://mlsanity.com/paper/2405.03427/) (2024)
- [A Variational Bayesian Inference Theory of Elasticity and Its Mixed Probabilistic Finite Element Method for Inverse Deformation Solutions in Any Dimension](https://mlsanity.com/paper/2410.07605/) (2024)
- [Predicting Stress in Two-phase Random Materials and Super-Resolution Method for Stress Images by Embedding Physical Information](https://mlsanity.com/paper/2504.18854/) (2025)
- [PINNs Algorithmic Framework for Simulation of Nonlinear Burgers' Type Models](https://mlsanity.com/paper/2506.12922/) (2025)

## Abstract (arXiv)

One of the obstacles hindering the scaling-up of the initial successes of machine learning in practical engineering applications is the dependence of the accuracy on the size of the database that "drives" the algorithms. Incorporating the already-known physical laws into the training process can significantly reduce the size of the required database. In this study, we establish a neural network-based computational framework to characterize the finite deformation of elastic plates, which in classic theories is described by the F"oppl--von K'arm'an (FvK) equations with a set of boundary conditions (BCs). A neural network is constructed by taking the spatial coordinates as the input and the displacement field as the output to approximate the exact solution of the FvK equations. The physical information (PDEs, BCs, and potential energies) is then incorporated into the loss function, and a pseudo dataset is sampled without knowing the exact solution to finally train the neural network. The prediction accuracy of the modeling framework is carefully examined by applying it to four different loading cases: in-plane tension with non-uniformly distributed stretching forces, in-plane central-hole tension, out-of-plane deflection, and buckling under compression. hlThree ways of formulating the loss function are compared: 1) purely data-driven, 2) PDE-based, and 3) energy-based. Through the comparison with the finite element simulations, it is found that all the three approaches can characterize the elastic deformation of plates with a satisfactory accuracy if trained properly. Compared with incorporating the PDEs and BCs in the loss, using the total potential energy shows certain advantage in terms of the simplicity of hyperparameter tuning and the computational efficiency.