# A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity

arXiv: https://arxiv.org/abs/2108.07243 · HTML: https://mlsanity.com/paper/2108.07243/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 71 references that are in the map.
- First posted on arXiv 16 Aug 2021.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Speech Recognition with Deep Recurrent Neural Networks](https://mlsanity.com/paper/1303.5778/) (2013)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Recent Advances in Convolutional Neural Networks](https://mlsanity.com/paper/1512.07108/) (2015)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Deep Learning for Medical Image Processing: Overview, Challenges and Future](https://mlsanity.com/paper/1704.06825/) (2017)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [PCANet: A Simple Deep Learning Baseline for Image Classification?](https://mlsanity.com/paper/1404.3606/) (2014)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [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)
- [A nonlocal physics-informed deep learning framework using the peridynamic differential operator](https://mlsanity.com/paper/2006.00446/) (2020)
- [Deep Generative Markov State Models](https://mlsanity.com/paper/1805.07601/) (2018)
- [Conditional deep surrogate models for stochastic, high-dimensional, and multi-fidelity systems](https://mlsanity.com/paper/1901.04878/) (2019)
- [Energy-based error bound of physics-informed neural network solutions in elasticity](https://mlsanity.com/paper/2010.09088/) (2020)
- [An accelerated hybrid data-driven/model-based approach for poroelasticity problems with multi-fidelity multi-physics data](https://mlsanity.com/paper/2012.00165/) (2020)
- [Parametric Gaussian Process Regression for Big Data](https://mlsanity.com/paper/1704.03144/) (2017)
- [ANN-aided incremental multiscale-remodelling-based finite strain poroelasticity](https://mlsanity.com/paper/2103.06569/) (2021)

## Built on by (most cited first, 12 of 12)

- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [A physics-informed deep neural network for surrogate modeling in classical elasto-plasticity](https://mlsanity.com/paper/2204.12088/) (2022)
- [Theory and implementation of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2311.06380/) (2023)
- [Physics Informed Neural Network for Dynamic Stress Prediction](https://mlsanity.com/paper/2211.16190/) (2022)
- [Reduced-PINN: An Integration-Based Physics-Informed Neural Networks for Stiff ODEs](https://mlsanity.com/paper/2208.12045/) (2022)
- [Aligning Perception, Reasoning, Modeling and Interaction: A Survey on Physical AI](https://mlsanity.com/paper/2510.04978/) (2025)
- [Physics-aware deep learning framework for linear elasticity](https://mlsanity.com/paper/2302.09668/) (2023)
- [A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials](https://mlsanity.com/paper/2605.31231/) (2026)
- [Stochastic Inference of Plate Bending from Heterogeneous Data: Physics-informed Gaussian Processes via Kirchhoff-Love Theory](https://mlsanity.com/paper/2405.12802/) (2024)
- [Collapsing Taylor Mode Automatic Differentiation](https://mlsanity.com/paper/2505.13644/) (2025)
- [Breaking the Precision Ceiling in Physics-Informed Neural Networks: A Hybrid Fourier-Neural Architecture for Ultra-High Accuracy](https://mlsanity.com/paper/2507.20929/) (2025)

## Abstract (arXiv)

We explore an application of the Physics Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions to a few reference biharmonic problems of elasticity and elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that are challenging to solve using classical numerical methods, and have not been addressed using PINNs. Our work highlights a novel application of classical analytical methods to guide the construction of efficient neural networks with the minimal number of parameters that are very accurate and fast to evaluate. In particular, we find that enriching feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.