On Parameter Identification in Three-Dimensional Elasticity and Discretisation with Physics-Informed Neural Networks

TL;DR

This paper develops a physics-informed neural network approach for identifying spatially varying parameters in 3D elasticity, providing stability guarantees and comparing neural network discretisation with traditional methods.

Contribution

It introduces a stable all-at-once optimisation framework for inverse elasticity problems and proves discretisation-independent stability estimates.

Findings

Neural network discretisation performs comparably to mesh-based methods.

Theoretical stability estimates are independent of discretisation.

Numerical examples validate the proposed approach.

Abstract

Physics-informed neural networks have emerged as a powerful tool in the scientific machine learning community, with applications to both forward and inverse problems. While they have shown considerable empirical success, significant challenges remain -- particularly regarding training stability and the lack of rigorous theoretical guarantees, especially when compared to classical mesh-based methods. In this work, we focus on the inverse problem of identifying a spatially varying parameter in a constitutive model of three-dimensional elasticity, using measurements of the system's state. This setting is especially relevant for non-invasive diagnosis in cardiac biomechanics, where one must also carefully account for the type of boundary data available. To address this inverse problem, we adopt an all-at-once optimisation framework, simultaneously estimating the state and parameter through a least-squares loss that encodes both available data and the governing physics. For this formulation, we prove stability estimates ensuring that our approach yields a stable approximation of the underlying ground-truth parameter of the physical system independent of a specific discretisation. We then proceed with a neural network-based discretisation and compare it to traditional mesh-based approaches. Our theoretical findings are complemented by illustrative numerical examples.