PROBABILISTIC PHYSICS-INFORMED NEURAL NETWORKS FOR ESTIMATING HETEROGENEOUS ELASTIC PROPERTIES FROM LOW-RESOLUTION AND NOISY DISPLACEMENT DATA

Estimating spatially heterogeneous elastic properties (Young's modulus and Poisson's ratio) from displacement data is a classic inverse problem. In practice, these measurements are often low-resolution and noisy, which amplifies errors and makes traditional physics-informed neural networks (PINNs) unstable. This paper addresses the challenge of recovering accurate elasticity maps under these degraded conditions without relying on manually tuned loss weights.

The authors introduce the Probabilistic Inverse Elasticity Physics-Informed Neural Network (PIE-PINN). Unlike deterministic PINNs, PIE-PINN treats displacement observations, strain discrepancies, and equilibrium residuals as probabilistic variables modeled by Laplace distributions. To handle low-resolution data, the framework employs a hybrid architecture: a B-spline network captures global displacement trends, while a standard neural network learns local variations. A hierarchical half-Cauchy model is used to adaptively adjust the scales of these residuals, effectively acting as an automated, principled weighting mechanism that downweights severe errors during training. The model is optimized using an alternating maximum-likelihood strategy that iteratively updates the displacement field and the residual scales.

PIE-PINN demonstrates superior robustness compared to existing deterministic PINN approaches, particularly when dealing with high noise levels and sparse, low-resolution observations. The B-spline-guided architecture effectively mitigates interpolation errors that typically plague numerical differentiation in low-resolution settings. By jointly estimating the mean displacement field and the adaptive loss weights, the framework avoids the instability associated with empirical weight selection, leading to more accurate and reliable recovery of elastic property distributions.