[Paper Review] A deep learning framework for solution and discovery in solid mechanics: linear elasticity.

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

Model Reduction and Neural Networks29 references21 citations

TL;DR

This paper introduces a multi-network Physics-Informed Neural Network (PINN) framework for solving and discovering solutions in linear elasticity, embedding momentum balance and constitutive laws directly into the loss function. The approach achieves superior accuracy and convergence using Isogeometric Analysis (IGA) and enables robust generalization across unseen parameters, demonstrating strong potential for surrogate modeling and sensitivity analysis.

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

• To develop a deep learning framework that integrates physical laws into neural networks for solid mechanics problems.
• To improve solution accuracy and convergence in linear elasticity using Isogeometric Analysis (IGA) within a PINN framework.
• To enable robust generalization across unseen material and loading parameters through physics-informed training.
• To explore transfer learning advantages in retraining with minimal data.
• To demonstrate the framework's applicability to surrogate modeling and sensitivity analysis.

Proposed method

• The framework employs a multi-network architecture to better represent field variables such as displacement and stress, improving accuracy over single-network PINNs.
• Momentum balance and linear elastic constitutive relations are embedded into the loss function using weak forms of the governing PDEs.
• Isogeometric Analysis (IGA) is used to parameterize the solution space, enhancing smoothness and convergence compared to low-order FEM.
• The model is trained on synthetic data generated from analytical and numerical reference solutions to validate accuracy and convergence.
• Transfer learning is applied by retraining on new parameter sets, leveraging previously learned physics to accelerate convergence.
• The framework enforces physical consistency by minimizing residual violations of the strong or weak forms of the PDEs during training.

Experimental results

Research questions

• RQ1 Can a multi-network PINN framework achieve higher accuracy in solving linear elasticity problems than single-network PINNs?
• RQ2 How does Isogeometric Analysis (IGA) compare to classic low-order FEM in terms of convergence and accuracy within a PINN framework?
• RQ3 To what extent can a PINN model trained on a limited set of parameters generalize to new, unseen parameter values?
• RQ4 Can the physics-informed framework support efficient transfer learning with accelerated retraining on new problems?
• RQ5 Can the framework serve as an effective surrogate model for sensitivity analysis in solid mechanics?

Key findings

• The multi-network PINN framework achieves improved representation of field variables compared to single-network PINNs, resulting in higher solution accuracy.
• IGA-based PINNs demonstrate superior convergence characteristics and accuracy compared to low-order FEM-based PINNs.
• The model exhibits strong generalization: trained on few parameters, it accurately predicts solutions for a wide range of new, unseen parameters.
• Transfer learning leads to vastly accelerated convergence during retraining, highlighting the framework's efficiency for parametric studies.
• Physics-informed training enhances robustness, enabling reliable solution discovery even with limited training data.
• The framework is applicable to surrogate modeling and sensitivity analysis due to its ability to generalize across parameter variations.