[Paper Review] A deep learning framework for solution and discovery in solid mechanics

Ehsan Haghighat, Maziar Raissi+3 more|arXiv (Cornell University)|Feb 14, 2020

Machine Learning in Materials Science35 references79 citations

TL;DR

The paper develops Physics-Informed Neural Networks (PINN) for solving and identifying parameters in solid mechanics, extends to nonlinear elastoplasticity, compares data sources (FEM/IGA), and demonstrates transfer learning and sensitivity analysis reductions.

ABSTRACT

We present the application of a class of deep learning, known as Physics Informed Neural Networks (PINN), to learning and discovery in solid mechanics. We explain how to incorporate the momentum balance and constitutive relations into PINN, and explore in detail the application to linear elasticity, and illustrate its extension to nonlinear problems through an example that showcases von~Mises elastoplasticity. While common PINN algorithms are based on training one deep neural network (DNN), we propose a multi-network model that results in more accurate representation of the field variables. To validate the model, we test the framework on synthetic data generated from analytical and numerical reference solutions. We study convergence of the PINN model, and show that Isogeometric Analysis (IGA) results in superior accuracy and convergence characteristics compared with classic low-order Finite Element Method (FEM). We also show the applicability of the framework for transfer learning, and find vastly accelerated convergence during network re-training. Finally, we find that honoring the physics leads to improved robustness: when trained only on a few parameters, we find that the PINN model can accurately predict the solution for a wide range of parameters new to the network---thus pointing to an important application of this framework to sensitivity analysis and surrogate modeling.

Motivation & Objective

• Introduce PINN for solid mechanics by embedding momentum balance and constitutive relations into the neural network framework.
• Develop a multi-network PINN architecture to improve field representation for displacement and stress.
• Demonstrate parameter identification (model inversion) within PINN for linear elasticity and elastoplasticity.
• Validate PINN on data from analytical solutions, FEM, and Isogeometric Analysis (IGA); compare accuracy and convergence.
• Explore transfer learning and sensitivity analysis using PINN as surrogate models.

Proposed method

• Formulate governing equations of linear elasticity and elastoplasticity within a PINN loss function that includes data misfit and physics residuals.
• Use five independent neural networks for ux, uy, sigma_xx, sigma_yy, sigma_xy to capture cross-dependencies via automatic differentiation.
• Train with data either from exact solutions or high-fidelity simulations, and treat material parameters (lambda, mu, sigma_Y) as trainable in identification mode.
• Compare data sources (FEM of varying element orders and IGA) to assess impact on convergence and accuracy.
• Demonstrate transfer learning by re-training a pre-trained PINN on new parameter datasets to accelerate convergence.
• Extend the framework to elastoplasticity using von Mises yield criterion and KKT conditions via penalty terms in the loss.

Experimental results

Research questions

• RQ1 Can PINNs accurately solve linear elasticity problems by enforcing momentum balance and constitutive laws?
• RQ2 How does a multi-network PINN compare to a single-network approach for representing field variables in solid mechanics?
• RQ3 Can PINNs identify material parameters (lambda, mu, yield stress) from data, and how robust is this to data source quality?
• RQ4 What is the impact of data source (analytical, FEM, or IGA) on convergence and accuracy of PINN training?
• RQ5 Is transfer learning feasible in PINNs for rapid adaptation to new material parameters and data distributions?

Key findings

• Independent networks for each variable (ux, uy, sigma_xx, sigma_yy, sigma_xy) yield more accurate parameter identification than a single network.
• Training on higher-order or more accurate data (IGA/FEM with higher continuity) improves convergence and accuracy of the PINN.
• Force-complete data (displacements, derivatives, and derived body forces) supports faster convergence than stress-complete data in parameter identification.
• PINN-trained models can robustly extrapolate to new parameter ranges, enabling effective sensitivity analysis with sparse data.
• Transfer learning substantially reduces retraining epochs when adapting to new datasets or parameter values.