[Paper Review] Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imaging

Enrui Zhang, Minglang Yin+1 more|arXiv (Cornell University)|Sep 2, 2020

Model Reduction and Neural NetworksPhysics and Astronomy19 references72 citations

TL;DR

The paper extends PINNs to identify spatially varying material properties in elasticity imaging by using two neural networks—one for the displacement field and one for the nonhomogeneous shear modulus—validated on a prototypical plane strain problem for incompressible Neo-Hookean tissue.

ABSTRACT

We apply Physics-Informed Neural Networks (PINNs) for solving identification problems of nonhomogeneous materials. We focus on the problem with a background in elasticity imaging, where one seeks to identify the nonhomogeneous mechanical properties of soft tissue based on the full-field displacement measurements under quasi-static loading. In our model, we apply two independent neural networks, one for approximating the solution of the corresponding forward problem, and the other for approximating the unknown material parameter field. As a proof of concept, we validate our model on a prototypical plane strain problem for incompressible hyperelastic tissue. The results show that the PINNs are effective in accurately recovering the unknown distribution of mechanical properties. By employing two neural networks in our model, we extend the capability of material identification of PINNs to include nonhomogeneous material parameter fields, which enables more flexibility of PINNs in representing complex material properties.

Motivation & Objective

• Motivate and leverage PINNs to solve inverse problems for nonhomogeneous hyperelastic solids in elasticity imaging.
• Develop a two-network PINN architecture to simultaneously approximate the forward solution and the spatially varying material parameter field.
• Validate the approach on a prototypical plane strain problem with incompressible Neo-Hookean tissue and assess accuracy of modulus recovery.

Proposed method

• Formulate a PINN with two networks: Net U to approximate the forward displacement and pressure fields, and Net μ to approximate the spatially varying shear modulus μ(X1,X2).
• Compute deformation gradient F and PK stress P using hat{P}_{iJ} = -hat{p} hat{F}_{iJ}^{-T} + hat{μ} hat{F}_{iJ} under plane strain and incompressibility.
• Enforce physics via PDE residuals for equilibrium, incompressibility constraint det(F)=1, and boundary conditions (Dirichlet and Neumann) through a composite loss L.
• Incorporate measured displacement data at N_u points and collocation points for PDEs and boundary conditions to train the networks.
• Train by minimizing L with respect to θ_U and θ_μ, enabling prediction of μ̃(X1,X2) = Net μ(X1,X2; θ̃_μ).
• Use a prototypical square domain, with a nonhomogeneous μ*(X1,X2) composed of a low-frequency background and a localized high-modulus region to mimic healthy and diseased tissue.

Experimental results

Research questions

• RQ1 Can PINNs identify a spatially varying shear modulus field from displacement data in a nonhomogeneous hyperelastic material?
• RQ2 Does a dual-network PINN architecture improve the recovery of nonhomogeneous material properties in elasticity imaging under quasi-static loading?
• RQ3 How accurately can the shear modulus field μ(X1,X2) be recovered in a plane strain incompressible Neo-Hookean model using PINNs?
• RQ4 What is the effect of data and physics terms on the convergence and accuracy of the inferred modulus field?

Key findings

• The PINN accurately recovers the spatial distribution of the shear modulus with local absolute errors below 0.01 in the test, while μ* ranges from about 0.15 to 0.37.
• Overall relative L2 error of the modulus field after training is about 1%.
• The total loss decreases from around 1e-1 to around 1e-4 during training, indicating satisfactory satisfaction of PDEs, BCs, incompressibility, and data terms.
• The dual-network approach extends PINNs to handle spatially varying material parameters, enabling more flexible material property representation in elasticity imaging.