# Probabilistic Physics-Informed Neural Networks for Estimating Heterogeneous Elastic Properties from Low-Resolution and Noisy Displacement Data

arXiv: https://arxiv.org/abs/2607.14563 · HTML: https://mlsanity.com/paper/2607.14563/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 65 references that are in the map.
- First posted on arXiv 16 Jul 2026.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (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)
- [Unsupervised discovery of interpretable hyperelastic constitutive laws](https://mlsanity.com/paper/2010.13496/) (2020)
- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [Calibrating constitutive models with full-field data via physics informed neural networks](https://mlsanity.com/paper/2203.16577/) (2022)
- [Self-adaptive weighting and sampling for physics-informed neural networks](https://mlsanity.com/paper/2511.05452/) (2025)
- [Physics-informed UNets for Discovering Hidden Elasticity in Heterogeneous Materials](https://mlsanity.com/paper/2306.01204/) (2023)
- [Robust Physics-Informed Neural Network Approach for Estimating Heterogeneous Elastic Properties from Noisy Displacement Data](https://mlsanity.com/paper/2506.14036/) (2025)

## Abstract (arXiv)

Estimating spatially heterogeneous elastic properties from low-resolution displacement measurements is a severely ill-posed inverse elasticity problem because low resolution obscures spatial details needed to distinguish heterogeneous property variations, and small measurement perturbations or fitting errors are amplified through inverse estimation. Existing inverse methods often rely on high-fidelity observations and manually prespecified loss weights, limiting their adaptability and making them sensitive to noise and resolution degradation. We propose a Probabilistic Inverse Elasticity Physics-Informed Neural Network (PIE-PINN) framework for robust estimation of Young's modulus and Poisson's ratio from noisy, low-resolution displacement data. PIE-PINN models displacement observation, strain-discrepancy, and equilibrium residuals using Laplace distributions within a unified probabilistic model. To improve robustness, the framework combines a B-spline-guided displacement network with a hierarchical half-Cauchy model for displacement residual scales. The B-spline provides a smooth global representation of the displacement field, while the neural network correction captures local variations. The hierarchical scale model adaptively downweights severe displacement fitting errors, enabling more robust recovery of the latent mean displacement field. An alternating maximum-likelihood training strategy updates the mean through weighted residual minimization and updates the scales to adjust the loss weights. Systematic case studies across varying noise levels and observation resolutions demonstrate the robustness of PIE-PINN.