# Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method

arXiv: https://arxiv.org/abs/2505.04875 · HTML: https://mlsanity.com/paper/2505.04875/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 78 references that are in the map.
- First posted on arXiv 8 May 2025.

## Builds on (most cited first, 29 of 29)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [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)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [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)
- [PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries](https://mlsanity.com/paper/2004.06490/) (2020)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs](https://mlsanity.com/paper/1911.02109/) (2019)
- [Stress field prediction in fiber-reinforced composite materials using a deep learning approach](https://mlsanity.com/paper/2111.05271/) (2021)
- [A deep learning energy method for hyperelasticity and viscoelasticity](https://mlsanity.com/paper/2201.08690/) (2022)
- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [Physics-Informed CoKriging: A Gaussian-Process-Regression-Based Multifidelity Method for Data-Model Convergence](https://mlsanity.com/paper/1811.09757/) (2018)
- [Operator Learning for Reconstructing Flow Fields from Sparse Measurements: an Energy Transformer Approach](https://mlsanity.com/paper/2501.08339/) (2025)
- [Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems](https://mlsanity.com/paper/2108.00037/) (2021)
- [Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Data](https://mlsanity.com/paper/2406.02581/) (2024)
- [Data assimilation and parameter identification for water waves using the nonlinear Schr"odinger equation and physics-informed neural networks](https://mlsanity.com/paper/2401.03708/) (2024)

## Built on by (most cited first, 1 of 1)

- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)

## Abstract (arXiv)

One use case of ``physics-informed neural networks'' (PINNs) is solution reconstruction, which aims to estimate the full-field state of a physical system from sparse measurements. Parameterized governing equations of the system are used in tandem with the measurements to regularize the regression problem. However, in real-world solution reconstruction problems, the parameterized governing equation may be inconsistent with the physical phenomena that give rise to the measurement data. We show that due to assuming consistency between the true and parameterized physics, PINNs-based approaches may fail to satisfy three basic criteria of interpretability, robustness, and data consistency. As we argue, these criteria ensure that (i) the quality of the reconstruction can be assessed, (ii) the reconstruction does not depend strongly on the choice of physics loss, and (iii) that in certain situations, the physics parameters can be uniquely recovered. In the context of elasticity and heat transfer, we demonstrate how standard formulations of the physics loss and techniques for constraining the solution to respect the measurement data lead to different ``constraint forces" -- which we define as additional source terms arising from the constraints -- and that these constraint forces can significantly influence the reconstructed solution. To avoid the potentially substantial influence of the choice of physics loss and method of constraint enforcement on the reconstructed solution, we propose the ``explicit constraint force method'' (ECFM) to gain control of the source term introduced by the constraint. We then show that by satisfying the criteria of interpretability, robustness, and data consistency, this approach leads to more predictable and customizable reconstructions from noisy measurement data, even when the parameterization of the missing physics is inconsistent with the measured system.