# Physics-informed neural networks (PINNs) for numerical model error approximation and superresolution

arXiv: https://arxiv.org/abs/2411.09728 · HTML: https://mlsanity.com/paper/2411.09728/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 50 references that are in the map.
- First posted on arXiv 14 Nov 2024.

## Builds on (most cited first, 8 of 8)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [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 deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials](https://mlsanity.com/paper/1807.09829/) (2018)
- [Inverse Problems with Learned Forward Operators](https://mlsanity.com/paper/2311.12528/) (2023)
- [Model reduction in acoustic inversion by artificial neural network](https://mlsanity.com/paper/2105.02225/) (2021)
- [Model-based reconstructions for quantitative imaging in photoacoustic tomography](https://mlsanity.com/paper/2311.15735/) (2023)

## Abstract (arXiv)

Numerical modeling errors are unavoidable in finite element analysis. The presence of model errors inherently reflects both model accuracy and uncertainty. To date there have been few methods for explicitly quantifying errors at points of interest (e.g. at finite element nodes). The lack of explicit model error approximators has been addressed recently with the emergence of machine learning (ML), which closes the loop between numerical model features/solutions and explicit model error approximations. In this paper, we propose physics-informed neural networks (PINNs) for simultaneous numerical model error approximation and superresolution. To test our approach, numerical data was generated using finite element simulations on a two-dimensional elastic plate with a central opening. Four- and eight-node quadrilateral elements were used in the discretization to represent the reduced-order and higher-order models, respectively. It was found that the developed PINNs effectively predict model errors in both x and y displacement fields with small differences between predictions and ground truth. Our findings demonstrate that the integration of physics-informed loss functions enables neural networks (NNs) to surpass a purely data-driven approach for approximating model errors.