mlsanityData to 7 Oct 2026

Polyconvex anisotropic hyperelasticity with neural networks

arXiv 2106.14623 ↗First posted 20 Jun 2021cond-mat.mtrl-sci cs.CE cs.LGcited by 54

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Abstract (arXiv)

In the present work, two machine learning based constitutive models for finite deformations are proposed. Using input convex neural networks, the models are hyperelastic, anisotropic and fulfill the polyconvexity condition, which implies ellipticity and thus ensures material stability. The first constitutive model is based on a set of polyconvex, anisotropic and objective invariants. The second approach is formulated in terms of the deformation gradient, its cofactor and determinant, uses group symmetrization to fulfill the material symmetry condition, and data augmentation to fulfill objectivity approximately. The extension of the dataset for the data augmentation approach is based on mechanical considerations and does not require additional experimental or simulation data. The models are calibrated with highly challenging simulation data of cubic lattice metamaterials, including finite deformations and lattice instabilities. A moderate amount of calibration data is used, based on deformations which are commonly applied in experimental investigations. While the invariant-based model shows drawbacks for several deformation modes, the model based on the deformation gradient alone is able to reproduce and predict the effective material behavior very well and exhibits excellent generalization capabilities. In addition, the models are calibrated with transversely isotropic data, generated with an analytical polyconvex potential. For this case, both models show excellent results, demonstrating the straightforward applicability of the polyconvex neural network constitutive models to other symmetry groups.

Key facts. Cited by 54 papers in the map (data through 7 Oct 2026). Builds on 12 of its 88 references that are in the map. First posted on arXiv 20 Jun 2021.

Builds on

12 of 88 references.

References · 12

  1. 2017A Unified Approach to Interpreting Model Predictionscited by 6,488
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  4. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
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  6. 2020Unsupervised discovery of interpretable hyperelastic constitutive lawscited by 51
  7. 2020Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticitycited by 49
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 7; 2023: 11; 2024: 11; 2025: 12; 2026: 12.

54 papers.

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  1. 2023Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamicscited by 56
  2. 2022A new family of Constitutive Artificial Neural Networks towards automated model discoverycited by 48
  3. 2024A review on data-driven constitutive laws for solidscited by 35
  4. 2022NN-EUCLID: deep-learning hyperelasticity without stress datacited by 30
  5. 2024Neural Material Adaptor for Visual Grounding of Intrinsic Dynamicscited by 30
  6. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  7. 2024Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responsescited by 18
  8. 2022Modular machine learning-based elastoplasticity: generalization in the context of limited datacited by 16
  9. 2023Data-driven anisotropic finite viscoelasticity using neural ordinary differential equationscited by 16
  10. 2023Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanicscited by 16
  11. 2023On sparse regression, Lp-regularization, and automated model discoverycited by 15
  12. 2022Calibrating constitutive models with full-field data via physics informed neural networkscited by 12
38 more
  1. 2025Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)cited by 12
  2. 2022Geometric deep learning for computational mechanics Part II: Graph embedding for interpretable multiscale plasticitycited by 11
  3. 2025UniPhy: Learning a Unified Constitutive Model for Inverse Physics Simulationcited by 11
  4. 2022Evolution TANN and the identification of internal variables and evolution equations in solid mechanicscited by 9
  5. 2023Stress representations for tensor basis neural networks: alternative formulations to Finger-Rivlin-Ericksencited by 9
  6. 2024EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomialscited by 9
  7. 2023Generative Hyperelasticity with Physics-Informed Probabilistic Diffusion Fieldscited by 7
  8. 2023Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINNcited by 6
  9. 2024Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networkscited by 6
  10. 2025Polyconvex Physics-Augmented Neural Network Constitutive Models in Principal Stretchescited by 5
  11. 2023Symmetry-enforcing neural networks with applications to constitutive modelingcited by 4
  12. 2024Improving the performance of Stein variational inference through extreme sparsification of physically-constrained neural network modelscited by 4
  13. 2025A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strainscited by 4
  14. 2022Neural Network Layers for Prediction of Positive Definite Elastic Stiffness Tensorscited by 3
  15. 2024Neural Networks for Generating Better Local Optima in Topology Optimizationcited by 3
  16. 2024Consistent machine learning for topology optimization with microstructure-dependent neural network material modelscited by 3
  17. 2025Input convex neural networks: universal approximation theorem and implementation for isotropic polyconvex hyperelastic energiescited by 3
  18. 2025Automated Constitutive Model Discovery by Pairing Sparse Regression Algorithms with Model Selection Criteriacited by 3
  19. 2026Towards Rapid Constitutive Model Discovery from Multi-Modal Data: Physics Augmented Finite Element Model Updating (paFEMU)cited by 3
  20. 2024Similarity Equivariant Graph Neural Networks for Homogenization of Metamaterialscited by 2
  21. 2025Input Specific Neural Networkscited by 2
  22. 2025COMMET: orders-of-magnitude speed-up in finite element method via batch-vectorized neural constitutive updatescited by 2
  23. 2026Hyperelastic constitutive model discovery with differentiable finite elements and structure-preserving neural networkscited by 2
  24. 2021MD-inferred neural network monoclinic finite-strain hyperelasticity models for β-HMX: Sobolev training and validation against physical constraintscited by 1
  25. 2023Exploring hyperelastic material model discovery for human brain cortex: multivariate analysis vs. artificial neural network approachescited by 1
  26. 2023Incorporating sufficient physical information into artificial neural networks: a guaranteed improvement via physics-based Rao-Blackwellizationcited by 1
  27. 2025Learning 3D Garment Animation from Trajectories of A Piece of Clothcited by 1
  28. 2025Differentiable neural network representation of multi-well, locally-convex potentialscited by 1
  29. 2026A Convex Route to Thermoelasticity: Learning Internal Energy and Dissipationcited by 1
  30. 2024Equivariant graph convolutional neural networks for the representation of homogenized anisotropic microstructural mechanical responsecited by 0
  31. 2024Learning Physics-Consistent Material Behavior from Dynamic Displacementscited by 0
  32. 2025Physics-based machine learning for fatigue lifetime prediction under non-uniform loading scenarioscited by 0
  33. 2025Hybrid twinning using PBDW and DeepONet for the effective state estimation and prediction on partially known systemscited by 0
  34. 2026Physics-Consistent Neural Networks for Learning Deformation and Director Fields in Microstructured Media with Loss-Based Validation Criteriacited by 0
  35. 2026LLM-driven design of physics-constrained constitutive models: two agents are better than onecited by 0
  36. 2026Finite Element-Based Material Learning via Automatic Differentiation: Learning constitutive neural network models from full-field deformation datacited by 0
  37. 2026CANN-EUCLID: unsupervised constitutive artificial neural network model discovery from full-field datacited by 0
  38. 2026Implementation of Hyperelastic Physics-Augmented Neural Networks in the Explicit Finite Element Codes Simcenter Radioss and OpenRadioss with Applications to Impact Eventscited by 0

50 most cited of 54 All 54 (JSON)