# Physics-Informed Neural Networks for Shell Structures

arXiv: https://arxiv.org/abs/2207.14291 · HTML: https://mlsanity.com/paper/2207.14291/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 39 references that are in the map.
- First posted on arXiv 26 Jul 2022.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches](https://mlsanity.com/paper/2010.06050/) (2020)
- [Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy](https://mlsanity.com/paper/2012.15744/) (2020)
- [Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates](https://mlsanity.com/paper/2010.05698/) (2020)

## Built on by (most cited first, 13 of 13)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data](https://mlsanity.com/paper/2305.10706/) (2023)
- [Recent Advances and Applications of Machine Learning in Experimental Solid Mechanics: A Review](https://mlsanity.com/paper/2303.07647/) (2023)
- [Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble](https://mlsanity.com/paper/2503.06320/) (2025)
- [PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations](https://mlsanity.com/paper/2412.05994/) (2024)
- [NeuralClothSim: Neural Deformation Fields Meet the Thin Shell Theory](https://mlsanity.com/paper/2308.12970/) (2023)
- [Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks](https://mlsanity.com/paper/2308.09605/) (2023)
- [Identifying Constitutive Parameters for Complex Hyperelastic Materials using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.15640/) (2023)
- [Finite-PINN: A Physics-Informed Neural Network with Finite Geometric Encoding for Solid Mechanics](https://mlsanity.com/paper/2412.09453/) (2024)
- [A Physics-informed Neural Network Approach for Robust Buckling Load Prediction and Reliability-Based Design of Thin Truncated Conical Shells](https://mlsanity.com/paper/2608.21818/) (2026)
- [GA-VINO: A Geometry-Aware Variational Physics-informed Neural Operator for Mindlin-Reissner Plates](https://mlsanity.com/paper/2606.16624/) (2026)
- [Continuous-Time Cloth Dynamics with Physics-Informed Neural ODEs](https://mlsanity.com/paper/2610.14342/) (2026)

## Abstract (arXiv)

The numerical modeling of thin shell structures is a challenge, which has been met by a variety of finite element (FE) and other formulations -- many of which give rise to new challenges, from complex implementations to artificial locking. As a potential alternative, we use machine learning and present a Physics-Informed Neural Network (PINN) to predict the small-strain response of arbitrarily curved shells. To this end, the shell midsurface is described by a chart, from which the mechanical fields are derived in a curvilinear coordinate frame by adopting Naghdi's shell theory. Unlike in typical PINN applications, the corresponding strong or weak form must therefore be solved in a non-Euclidean domain. We investigate the performance of the proposed PINN in three distinct scenarios, including the well-known Scordelis-Lo roof setting widely used to test FE shell elements against locking. Results show that the PINN can accurately identify the solution field in all three benchmarks if the equations are presented in their weak form, while it may fail to do so when using the strong form. In the thin-thickness limit, where classical methods are susceptible to locking, training time notably increases as the differences in scaling of the membrane, shear, and bending energies lead to adverse numerical stiffness in the gradient flow dynamics. Nevertheless, the PINN can accurately match the ground truth and performs well in the Scordelis-Lo roof benchmark, highlighting its potential for a drastically simplified alternative to designing locking-free shell FE formulations.