# Physics-aware deep learning framework for linear elasticity

arXiv: https://arxiv.org/abs/2302.09668 · HTML: https://mlsanity.com/paper/2302.09668/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 122 references that are in the map.
- First posted on arXiv 19 Feb 2023.

## Builds on (most cited first, 30 of 32)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Object Detection with Deep Learning: A Review](https://mlsanity.com/paper/1807.05511/) (2018)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [Machine learning for metal additive manufacturing: Predicting temperature and melt pool fluid dynamics using physics-informed neural networks](https://mlsanity.com/paper/2008.13547/) (2020)
- [Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imaging](https://mlsanity.com/paper/2009.04525/) (2020)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics](https://mlsanity.com/paper/2103.03174/) (2021)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Adaptive Self-supervision Algorithms for Physics-informed Neural Networks](https://mlsanity.com/paper/2207.04084/) (2022)
- [A nonlocal physics-informed deep learning framework using the peridynamic differential operator](https://mlsanity.com/paper/2006.00446/) (2020)
- [Energy-based error bound of physics-informed neural network solutions in elasticity](https://mlsanity.com/paper/2010.09088/) (2020)
- [How to Avoid Trivial Solutions in Physics-Informed Neural Networks](https://mlsanity.com/paper/2112.05620/) (2021)
- [Physics-Incorporated Convolutional Recurrent Neural Networks for Source Identification and Forecasting of Dynamical Systems](https://mlsanity.com/paper/2004.06243/) (2020)
- [A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity](https://mlsanity.com/paper/2108.07243/) (2021)
- [Physics-informed neural networks for modeling rate- and temperature-dependent plasticity](https://mlsanity.com/paper/2201.08363/) (2022)
- [A fast accurate fine-grain object detection model based on YOLOv4 deep neural network](https://mlsanity.com/paper/2111.00298/) (2021)
- [Precise Single-stage Detector](https://mlsanity.com/paper/2210.04252/) (2022)
- [Understanding EEG signals for subject-wise Definition of Armoni Activities](https://mlsanity.com/paper/2301.00948/) (2023)

## Built on by (most cited first, 2 of 2)

- [Physics-Regularized Multi-Modal Image Assimilation for Brain Tumor Localization](https://mlsanity.com/paper/2409.20409/) (2024)
- [A Computer Vision Enabled damage detection model with improved YOLOv5 based on Transformer Prediction Head](https://mlsanity.com/paper/2303.04275/) (2023)

## Abstract (arXiv)

The paper presents an efficient and robust data-driven deep learning (DL) computational framework developed for linear continuum elasticity problems. The methodology is based on the fundamentals of the Physics Informed Neural Networks (PINNs). For an accurate representation of the field variables, a multi-objective loss function is proposed. It consists of terms corresponding to the residual of the governing partial differential equations (PDE), constitutive relations derived from the governing physics, various boundary conditions, and data-driven physical knowledge fitting terms across randomly selected collocation points in the problem domain. To this end, multiple densely connected independent artificial neural networks (ANNs), each approximating a field variable, are trained to obtain accurate solutions. Several benchmark problems including the Airy solution to elasticity and the Kirchhoff-Love plate problem are solved. Performance in terms of accuracy and robustness illustrates the superiority of the current framework showing excellent agreement with analytical solutions. The present work combines the benefits of the classical methods depending on the physical information available in analytical relations with the superior capabilities of the DL techniques in the data-driven construction of lightweight, yet accurate and robust neural networks. The models developed herein can significantly boost computational speed using minimal network parameters with easy adaptability in different computational platforms.