Source-linked AI summary
NN-EUCLID: deep-learning hyperelasticity without stress data
Prakash Thakolkaran, Akshay Joshi, Yiwen Zheng, Moritz Flaschel, Laura De Lorenzis, Siddhant Kumar
TL;DR
Stress-label requirements limit constitutive-model learning because realistic experiments primarily provide displacements and global force measurements. NN-EUCLID uses a physics-motivated momentum-balance loss with input-convex neural networks to learn hyperelastic laws from these measurements. It produces physically admissible models that generalize beyond training strains and identify hidden anisotropic directions, while remaining subject to ill-posedness and current scope limitations.
Problem
Stress labels are difficult to obtain experimentally, while force measurements provide only boundary-aggregated stress projections and simple tests insufficiently probe high-dimensional stress-strain behavior.
Method
NN-EUCLID learns hyperelastic constitutive models from full-field displacements and global forces by minimizing a physics-motivated loss based on conservation of linear momentum, using input-convex neural networks.
Results
NN-EUCLID learns physically admissible isotropic and anisotropic models, generalizes beyond observed strain states, and accurately identifies unknown fiber orientations.
Takeaways & Limitations
The framework enables stress-free-data learning of constitutive models using realistically measurable displacement and force data, with deployment in finite-element simulations supported by the demonstrated generalization and accuracy.
Takeaways & Limitations
The inverse problem is highly ill-posed, admits multiple solutions and local minima, and the demonstrated formulation assumes plane strain.
Abstract
from arXiv · showhide
We propose a new approach for unsupervised learning of hyperelastic constitutive laws with physics-consistent deep neural networks. In contrast to supervised learning, which assumes the availability of stress-strain pairs, the approach only uses realistically measurable full-field displacement and global reaction force data, thus it lies within the scope of our recent framework for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID) and we denote it as NN-EUCLID. The absence of stress labels is compensated for by leveraging a physics-motivated loss function based on the conservation of linear momentum to guide the learning process. The constitutive model is based on input-convex neural networks, which are capable of learning a function that is convex with respect to its inputs. By employing a specially designed neural network architecture, multiple physical and thermodynamic constraints for hyperelastic constitutive laws, such as material frame indifference, (poly-)convexity, and stress-free reference configuration are automatically satisfied. We demonstrate the ability of the approach to accurately learn several hidden isotropic and anisotropic hyperelastic constitutive laws - including e.g., Mooney-Rivlin, Arruda-Boyce, Ogden, and Holzapfel models - without using stress data. For anisotropic hyperelasticity, the unknown anisotropic fiber directions are automatically discovered jointly with the constitutive model. The neural network-based constitutive models show good generalization capability beyond the strain states observed during training and are readily deployable in a general finite element framework for simulating complex mechanical boundary value problems with good accuracy.
1. Introduction
Conventional constitutive modeling and supervised data-driven methods are limited by model expressiveness and the need for extensive stress labels. NN-EUCLID addresses these limitations by learning physically admissible hyperelastic constitutive models from displacement and reaction-force data without stress data.
- Motivation: Stress labels are a major bottleneck because experiments typically provide only boundary projections of stress tensors through force measurements.Computational stress-label generation is costly and depends on lower-scale modeling, while simple experiments insufficiently probe high-dimensional stress-strain spaces.
- Motivation: EUCLID learns constitutive laws without stress data using full-field displacement and global reaction-force measurements.The framework uses sparse regression over candidate functions to produce interpretable models.
- Motivation: Neural networks offer substantially greater expressive power than sparse regression because they can represent generic constitutive behavior without a fixed feature library.This increased approximation power comes at the expense of interpretability.
- Contribution: NN-EUCLID is introduced as an unsupervised neural-network approach for learning isotropic and anisotropic hyperelastic constitutive models without stress data.The approach uses input-convex neural networks to represent the constitutive model.
- Contribution: The learned models are physically admissible, generalize beyond observed strain states, and can automatically discover unknown principal anisotropy directions.These capabilities are reported for the proposed unsupervised neural-network constitutive models.
2. Unsupervised deep learning of hyperelastic constitutive laws
NN-EUCLID learns hyperelastic constitutive laws from full-field displacement and reaction-force data by combining finite-element reconstruction, physics-consistent ICNNs, and momentum-balance residual minimization. Its architecture incorporates objectivity, convexity-related stability constraints, and a stress-free reference configuration while jointly learning unknown anisotropic fiber orientations.
- Problem setup: The method assumes a homogeneous hyperelastic specimen with potentially anisotropic, unknown fiber orientations and uses a complex geometry to generate diverse strain states.Fiber orientations are trainable parameters and are restricted to [0, π) because of two-fold symmetry.
- Field reconstruction: Point-wise displacement data and a finite-element mesh reconstruct continuous displacement and deformation-gradient fields from measured observations.The mesh uses linear triangular elements with a single quadrature point at each element barycenter.
- Constitutive model: The constitutive model derives stress and tangent responses from a strain-energy density represented by an ICNN operating on objective strain invariants.The architecture uses isotropic and anisotropic deviatoric invariants, volumetric quantities, trainable ICNN weights, and trainable fiber directions.
- Physical constraints: The architecture enforces objectivity through strain invariants, local convexity through the ICNN structure, and zero stress at the reference configuration through an in situ correction.Polyconvexity is used as a tractable relaxation of quasiconvexity, while the energy and stress corrections are updated during training.
- Unsupervised learning: Without energy or stress labels, the model parameters are learned by minimizing weak-form linear-momentum residuals computed from predicted stresses and measured forces.Residuals are minimized point-wise for free degrees of freedom and in aggregated form for constrained degrees of freedom with measured reactions.
- Optimization: Because the inverse problem is ill-posed and non-convex in the network parameters, independently initialized ICNN ensembles are filtered by their loss values.Accepted models are those with distinctively low losses; high-loss solutions trapped in poor local minima are rejected.
3. Numerical benchmarks
The numerical benchmarks train ICNN-based constitutive models on noisy synthetic displacement and reaction-force data, then evaluate their accuracy, generalization, and ability to recover hidden anisotropy.
- Data generation: Synthetic FEM data emulate DIC measurements from a square plate with a hole under displacement-controlled asymmetric biaxial tension.The data include nodal displacements and reaction forces, with artificial noise added to the displacement measurements.
- Data generation: Noise levels are σu = 10^-4 and σu = 10^-3, representing low- and high-noise cases relative to specimen length.Each noisy displacement snapshot is spatially denoised using kernel ridge regression before training.
- Evaluation protocol: ICNN models are evaluated on uniaxial, biaxial, simple-shear, and pure-shear deformation paths that do not contribute to training.The accepted models are selected from an ensemble of 30 independently initialized ICNNs using the final loss value.
- Model accuracy: The learned strain energies and first Piola-Kirchhoff stresses agree well with the ground-truth models across both noise levels.The evaluation covers all benchmark models and the six deformation paths, although larger extrapolative deformations can produce mismatches in strain energy density.
- Generalization and hidden anisotropy: Validation on a more complex specimen confirms generalization beyond training strain states, while the ICNNs accurately identify unknown fiber orientations under both noise levels.The validation geometry contains two asymmetric elliptical holes and is loaded in displacement-controlled uniaxial tension without related training data.
4. Conclusion and Outlook
NN-EUCLID learns isotropic and anisotropic hyperelastic constitutive behavior without stress data, using measurable displacement and force data with physics-consistent ICNNs. The reported benchmarks cover low- and high-noise settings across isotropic and anisotropic model families, including automatic discovery of anisotropic fiber orientations.
- Core contribution: NN-EUCLID learns isotropic and anisotropic hyperelastic constitutive behavior without relying on stress data.The approach uses full-field displacements and global forces, while a momentum-based loss guides unsupervised learning.
- Physical structure: ICNN-based constitutive models enforce material objectivity, local material stability, and a stress-free reference configuration through their architecture.
- Anisotropic discovery: For anisotropic hyperelasticity, the ICNN framework automatically discovers unknown principal directions of anisotropy, including fiber orientations.
- Benchmark evaluation: The benchmarks evaluate strain-energy predictions along multiple deformation paths for low-noise cases spanning NH, IH, HW, GT, AB, OG, AI45, AI60, and HZ models.The hidden true constitutive response is shown for reference in each benchmark group.
- Benchmark evaluation: The same benchmark families are also evaluated under high noise, including NH, IH, HW, GT, AB, OG, AI45, AI60, and HZ cases.The high-noise figures retain comparisons with the hidden true constitutive response.
Appendix A. Implementation details
The implementation appendix collects the parameters and hyperparameters used for data generation and neural-network training in Table A.1.
- Implementation details: Table A.1 lists the parameters and hyperparameters used for data generation and neural-network training.
Appendix A.1. Data generation
The data-generation procedure normalizes the specimen fields, denoises displacement measurements, and projects them onto a coarser mesh for data efficiency.
- Normalization and preprocessing: All lengths and displacements are normalized by the side length of the undeformed specimen.
- Mesh and preprocessing: The training specimen uses a high-resolution mesh with 63,601 nodes before displacement preprocessing.
- Mesh and preprocessing: Noisy displacements are spatially denoised with the KRR denoiser and then projected onto a coarser mesh with n_n = 1,441 nodes for data efficiency.
Appendix A.2. ICNN
The ICNN implementation uses a three-hidden-layer network with 64 neurons per layer and trains its parameters with Adam and automatic-differentiation backpropagation.
- Architecture: The ICNN has three hidden layers with 64 neurons each.The appendix specifies N = 4 and d_1 = d_2 = d_3 = 64.
- Optimization: Adam optimization with automatic-differentiation backpropagation trains the ICNN parameters Q and A.
- Optimization: Training runs for 500 epochs, which was observed to be sufficient for the loss to converge to an acceptable value.
Appendix A.3. Anisotropy
For anisotropic hyperelasticity, the model represents fiber orientations through trainable angle parameters while avoiding constrained optimization by reparameterizing them with unconstrained variables.
- The anisotropic model introduces angle parameters A to compute anisotropic invariants.
- Fiber orientations are parameterized by trainable variables ζ_i ∈ R instead of directly optimizing constrained angles α_i ∈ [0, π).
- This change of variables enables unconstrained optimization, which is easier than optimizing the bounded orientation angles directly.
Appendix B. Pseudocode for unsupervised ICNN training
Algorithm 1 summarizes the unsupervised training procedure for the ICNN-based constitutive models.
- Algorithm 1 provides the pseudocode for unsupervised training of the ICNN-based constitutive models.
Appendix C. Model accuracy in stress predictions
Stress predictions from the ICNN models are compared with ground-truth constitutive responses along six deformation paths under both noise levels.
- Figures C.12–C.17 compare ICNN-predicted first Piola–Kirchhoff stresses with ground truth along six deformation paths for both noise levels.
Appendix D. Constitutive models trained without input-convex architecture and smoothness
Replacing the constrained, smooth ICNN with an unconstrained feed-forward network produces non-smooth and oscillatory stress responses, limiting FEM deployment.
- The comparison replaces the ICNN with a feed-forward network lacking convexity constraints and using nonsmooth ReLU activations.
- Although strain-energy predictions are somewhat acceptable, the corresponding stress responses contain spurious, oscillatory, and non-smooth artifacts.
- The resulting unconstrained feed-forward network cannot be deployed in a finite-element framework.