[Paper Review] Neural networks meet anisotropic hyperelasticity: A framework based on generalized structure tensors and isotropic tensor functions

Karl A. Kalina, Jörg Brummund+2 more|arXiv (Cornell University)|Oct 4, 2024

Elasticity and Material Modeling4 citations

TL;DR

This paper proposes a physics-augmented neural network (PANN) framework for modeling anisotropic hyperelastic materials using generalized structure tensors and isotropic invariants. By constructing energy-based models from deformation invariants and enforcing physical principles via architecture and regularization, the method achieves high accuracy with minimal data, outperforming component-based networks in both interpolation and extrapolation, especially for complex microstructures like fiber-reinforced composites.

ABSTRACT

We present a data-driven framework for the multiscale modeling of anisotropic finite strain elasticity based on physics-augmented neural networks (PANNs). Our approach allows the efficient simulation of materials with complex underlying microstructures which reveal an overall anisotropic and nonlinear behavior on the macroscale. By using a set of invariants as input, an energy-type output and by adding several correction terms to the overall energy density functional, the model fulfills multiple physical principles by construction. The invariants are formed from the right Cauchy-Green deformation tensor and fully symmetric 2nd, 4th or 6th order structure tensors which enables to describe a wide range of symmetry groups. Besides the network parameters, the structure tensors are simultaneously calibrated during training so that the underlying anisotropy of the material is reproduced most accurately. In addition, sparsity of the model with respect to the number of invariants is enforced by adding a trainable gate layer and using lp regularization. Our approach works for data containing tuples of deformation, stress and material tangent, but also for data consisting only of tuples of deformation and stress, as is the case in real experiments. The developed approach is exemplarily applied to several representative examples, where necessary data for the training of the PANN surrogate model are collected via computational homogenization. We show that the proposed model achieves excellent interpolation and extrapolation behaviors. In addition, the approach is benchmarked against an NN model based on the components of the right Cauchy-Green deformation tensor.

Motivation & Objective

• To develop a data-driven framework for multiscale modeling of anisotropic finite strain elasticity in materials with complex microstructures.
• To ensure that the neural network model satisfies fundamental physical principles—such as objectivity, material symmetry, and thermodynamic consistency—by construction.
• To enable efficient learning from sparse experimental or simulation data, including only stress-deformation pairs, by leveraging invariant-based inputs and trainable structure tensors.
• To achieve model sparsity through a trainable gate layer with ℓp regularization, reducing the number of active invariants without sacrificing accuracy.
• To benchmark the performance against component-based neural networks, demonstrating superior generalization and robustness.

Proposed method

• The framework uses invariants derived from the right Cauchy-Green tensor and fully symmetric 2nd, 4th, or 6th-order structure tensors to encode material anisotropy.
• The neural network outputs the strain energy density as a function of these invariants, ensuring thermodynamic consistency and objectivity by design.
• A trainable gate layer with ℓp regularization (p = 1/4) enforces sparsity by identifying and suppressing irrelevant invariants during training.
• Structure tensors are simultaneously calibrated during training to accurately represent the underlying anisotropy of the material.
• The loss function combines stress and material tangent prediction losses (0.7σ + 0.3c) and includes a gate regularization term with tunable weight to balance sparsity and accuracy.
• The model is trained on data from computational homogenization of representative volume elements (RVEs), including cases with only deformation-stress pairs.

Experimental results

Research questions

• RQ1 Can a physics-augmented neural network model accurately capture complex anisotropic hyperelastic behavior using invariant-based inputs and structure tensors?
• RQ2 How does the inclusion of trainable structure tensors improve the model’s ability to represent material symmetry and anisotropy?
• RQ3 To what extent does the trainable gate layer with ℓp regularization reduce model complexity while maintaining predictive accuracy?
• RQ4 How does the invariant-based PANN compare to component-based neural networks in terms of interpolation and extrapolation performance?
• RQ5 Can the framework generalize effectively when trained on sparse data, such as only deformation and stress tuples?

Key findings

• The invariant-based PANN achieved losses below 1×10⁻⁴ for four of five RVEs with just two hidden layers of 16 neurons, while the coordinate-based model required three hidden layers and still failed to converge below 5×10⁻³.
• For the RVE with cubic spheres, the invariant-based model achieved a loss below 1×10⁻⁴ with two or more hidden layers, whereas the coordinate-based model showed no significant improvement beyond 64 neurons.
• The trainable gate layer with wgate = 5×10⁻⁵ successfully reduced the number of active invariants across all RVEs without degrading prediction performance.
• The model demonstrated excellent extrapolation behavior, maintaining high accuracy even when tested on deformation states outside the training distribution.
• The framework outperformed component-based networks in both accuracy and data efficiency, especially for materials with complex anisotropy such as fiber-reinforced composites.
• The use of generalized structure tensors enabled the model to represent a wide range of symmetry groups, including transverse isotropy and cubic symmetry, with high fidelity.