# Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity

arXiv: https://arxiv.org/abs/2001.04292 · HTML: https://mlsanity.com/paper/2001.04292/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jan 2020.

## Built on by (most cited first, 30 of 49)

- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics](https://mlsanity.com/paper/2304.14369/) (2023)
- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [NN-EUCLID: deep-learning hyperelasticity without stress data](https://mlsanity.com/paper/2205.06664/) (2022)
- [Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.03774/) (2021)
- [MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations](https://mlsanity.com/paper/2211.00713/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Large Language Model Agent as a Mechanical Designer](https://mlsanity.com/paper/2404.17525/) (2024)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Local approximate Gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks](https://mlsanity.com/paper/2105.04554/) (2021)
- [Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations](https://mlsanity.com/paper/2302.03598/) (2023)
- [Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)](https://mlsanity.com/paper/2503.05617/) (2025)
- [UniPhy: Learning a Unified Constitutive Model for Inverse Physics Simulation](https://mlsanity.com/paper/2505.16971/) (2025)
- [Equivariant geometric learning for digital rock physics: estimating formation factor and effective permeability tensors from Morse graph](https://mlsanity.com/paper/2104.05608/) (2021)
- [Geometric deep learning for computational mechanics Part II: Graph embedding for interpretable multiscale plasticity](https://mlsanity.com/paper/2208.00246/) (2022)
- [Data-driven discovery of interpretable causal relations for deep learning material laws with uncertainty propagation](https://mlsanity.com/paper/2105.09980/) (2021)
- [Discovering interpretable elastoplasticity models via the neural polynomial method enabled symbolic regressions](https://mlsanity.com/paper/2307.13149/) (2023)
- [Stress representations for tensor basis neural networks: alternative formulations to Finger-Rivlin-Ericksen](https://mlsanity.com/paper/2308.11080/) (2023)
- [Mesh-based graph convolutional neural networks for modeling materials with microstructure](https://mlsanity.com/paper/2107.00090/) (2021)
- [Learning the nonlinear dynamics of soft mechanical metamaterials with graph networks](https://mlsanity.com/paper/2202.13775/) (2022)
- [Generative Hyperelasticity with Physics-Informed Probabilistic Diffusion Fields](https://mlsanity.com/paper/2310.03745/) (2023)
- [A Microstructure-based Graph Neural Network for Accelerating Multiscale Simulations](https://mlsanity.com/paper/2402.13101/) (2024)
- [Predicting the Mechanical Properties of Biopolymer Gels Using Neural Networks Trained on Discrete Fiber Network Data](https://mlsanity.com/paper/2101.11712/) (2021)
- [Learning Mechanically Driven Emergent Behavior with Message Passing Neural Networks](https://mlsanity.com/paper/2202.01380/) (2022)
- [LS-DYNA Machine Learning-based Multiscale Method for Nonlinear Modeling of Short Fiber-Reinforced Composites](https://mlsanity.com/paper/2301.02738/) (2023)
- [A Large Language Model and Denoising Diffusion Framework for Targeted Design of Microstructures with Commands in Natural Language](https://mlsanity.com/paper/2409.14473/) (2024)
- [Cooperative data-driven modeling](https://mlsanity.com/paper/2211.12971/) (2022)
- [Convolution, aggregation and attention based deep neural networks for accelerating simulations in mechanics](https://mlsanity.com/paper/2212.01386/) (2022)

## Abstract (arXiv)

This paper is the first attempt to use geometric deep learning and Sobolev training to incorporate non-Euclidean microstructural data such that anisotropic hyperelastic material machine learning models can be trained in the finite deformation range. While traditional hyperelasticity models often incorporate homogenized measures of microstructural attributes, such as porosity averaged orientation of constitutes, these measures cannot reflect the topological structures of the attributes. We fill this knowledge gap by introducing the concept of weighted graph as a new mean to store topological information, such as the connectivity of anisotropic grains in assembles. Then, by leveraging a graph convolutional deep neural network architecture in the spectral domain, we introduce a mechanism to incorporate these non-Euclidean weighted graph data directly as input for training and for predicting the elastic responses of materials with complex microstructures. To ensure smoothness and prevent non-convexity of the trained stored energy functional, we introduce a Sobolev training technique for neural networks such that stress measure is obtained implicitly from taking directional derivatives of the trained energy functional. By optimizing the neural network to approximate both the energy functional output and the stress measure, we introduce a training procedure the improves efficiency and generalize the learned energy functional for different microstructures. The trained hybrid neural network model is then used to generate new stored energy functional for unseen microstructures in a parametric study to predict the influence of elastic anisotropy on the nucleation and propagation of fracture in the brittle regime.