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

2026-07-16

AI Summary

This work addresses the severely ill-posed inverse problem of inferring spatially heterogeneous elastic parameters—such as Young’s modulus and Poisson’s ratio—from low-resolution, noisy displacement data. Conventional approaches are highly sensitive to noise and rely on manually tuned loss weights. To overcome these limitations, the authors propose the PIE-PINN framework, which unifies displacement observations, strain deviations, and equilibrium residuals under a Laplace distribution assumption. By integrating a B-spline-guided displacement network with hierarchical half-Cauchy priors, the method enables adaptive loss weighting without manual intervention. This hybrid architecture leverages the global smoothness of B-splines and the local representational capacity of neural networks, significantly enhancing robustness and accuracy in parameter inversion under high-noise and low-resolution conditions, outperforming existing methods that require high-fidelity data.

Abstract

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.

Problem

• inverse elasticity
• heterogeneous elastic properties
• low-resolution data
• noisy displacement
• ill-posed problem

Innovation

• Probabilistic Physics-Informed Neural Networks
• Inverse elasticity
• Heterogeneous material properties
• B-spline displacement representation
• Hierarchical half-Cauchy model