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

TL;DR

This work develops a discretisation-invariant stability framework for inverse parameter identification in three-dimensional elasticity within a physics-informed neural network (PINN) setting. It establishes conditional stability transfers from PDEs to all-at-once optimization, analyzes both isotropic and anisotropic constitutive laws, and proves existence and convergence results in the presence of noise. The authors compare PINN discretisations to mesh-based finite-element approaches, deriving approximation/error bounds for neural networks and classical FE spaces, and demonstrate through numerical experiments that PINNs can achieve competitive reconstructions, particularly for smooth ground-truths, while FE methods face stability challenges in all-at-once formulations. The study emphasizes the potential of hybrid strategies and improved initialisations to combine the strengths of PINNs and FE methods for robust, high-dimensional parameter identification in cardiac biomechanics. Overall, the paper provides rigorous discretisation-invariant error estimates and practical insights into the trade-offs between PINN-based and FE-based approaches for inverse elasticity problems.

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.

Figures (3)

• Figure 1: Deformed configuration according to the generated ground-truth displacement $\mathbf u^*$ with different loads applied on the top surface.
• Figure 2: Box-plots of the relative error on $\epsilon_{\mu; \text{rel}}$ for each configuration. Each plot shows the results for a different noise level $\delta$. The x-axis indicates the width of the displacement network $N_\mathbf{u}$ and the color the width of the shear modulus network $N_\mu$.
• Figure 3: Comparison of the best reconstructed fields for the case $\delta=0.01$. The top left plot displays the PINNs reconstruction with $N_{\mathbf{u}} = 32$ and $N_\mu = 2$, with the color range clipped between $\SIrange{8}{16}{\kilo\pascal}$. The top right plot shows the least-squares FE approach with $N = 1$. The bottom left plot presents the weak formulation using the all-at-once approach for $N = 6$, while the bottom right plot illustrates the weak formulation using the reduced approach for $N = 6$. For the latter three plots, $\alpha$ is clipped between 1 and 2.

Theorems & Definitions (42)

• Remark 2.1
• Remark 2.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 3.1
• proof
• Theorem 3.1
• proof