# Energy-based error bound of physics-informed neural network solutions in elasticity

arXiv: https://arxiv.org/abs/2010.09088 · HTML: https://mlsanity.com/paper/2010.09088/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2020.

## Built on by (most cited first, 15 of 15)

- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [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)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [Preconditioning for Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.00531/) (2024)
- [A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity](https://mlsanity.com/paper/2108.07243/) (2021)
- [Non-intrusive reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation](https://mlsanity.com/paper/2101.11810/) (2021)
- [Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles](https://mlsanity.com/paper/2505.06459/) (2025)
- [Physics-aware deep learning framework for linear elasticity](https://mlsanity.com/paper/2302.09668/) (2023)
- [Stochastic Inference of Plate Bending from Heterogeneous Data: Physics-informed Gaussian Processes via Kirchhoff-Love Theory](https://mlsanity.com/paper/2405.12802/) (2024)
- [Astral: training physics-informed neural networks with error majorants](https://mlsanity.com/paper/2406.02645/) (2024)
- [Parsimonious Universal Function Approximator for Elastic and Elasto-Plastic Cavity Expansion Problems](https://mlsanity.com/paper/2407.19074/) (2024)
- [Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials](https://mlsanity.com/paper/2405.03427/) (2024)
- [A decomposition-based robust training of physics-informed neural networks for nearly incompressible linear elasticity](https://mlsanity.com/paper/2505.21994/) (2025)
- [Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds](https://mlsanity.com/paper/2606.12050/) (2026)

## Abstract (arXiv)

An energy-based a posteriori error bound is proposed for the physics-informed neural network solutions of elasticity problems. An admissible displacement-stress solution pair is obtained from a mixed form of physics-informed neural networks, and the proposed error bound is formulated as the constitutive relation error defined by the solution pair. Such an error estimator provides an upper bound of the global error of neural network discretization. The bounding property, as well as the asymptotic behavior of the physics-informed neural network solutions, are studied in a demonstrating example.