mlsanityData to 7 Oct 2026

Neural networks meet hyperelasticity: A guide to enforcing physics

arXiv 2302.02403 ↗First posted 5 Feb 2023cs.CEcited by 45

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Abstract (arXiv)

In the present work, a hyperelastic constitutive model based on neural networks is proposed which fulfills all common constitutive conditions by construction, and in particular, is applicable to compressible material behavior. Using different sets of invariants as inputs, a hyperelastic potential is formulated as a convex neural network, thus fulfilling symmetry of the stress tensor, objectivity, material symmetry, polyconvexity, and thermodynamic consistency. In addition, a physically sensible stress behavior of the model is ensured by using analytical growth terms, as well as normalization terms which ensure the undeformed state to be stress free and with zero energy. In particular, polyconvex, invariant-based stress normalization terms are formulated for both isotropic and transversely isotropic material behavior. By fulfilling all of these conditions in an exact way, the proposed physics-augmented model combines a sound mechanical basis with the extraordinary flexibility that neural networks offer. Thus, it harmonizes the theory of hyperelasticity developed in the last decades with the up-to-date techniques of machine learning. Furthermore, the non-negativity of the hyperelastic neural network-based potentials is numerically examined by sampling the space of admissible deformations states, which, to the best of the authors' knowledge, is the only possibility for the considered nonlinear compressible models. For the isotropic neural network model, the sampling space required for that is reduced by analytical considerations. In addition, a proof for the non-negativity of the compressible Neo-Hooke potential is presented. The applicability of the model is demonstrated by calibrating it on data generated with analytical potentials, which is followed by an application of the model to finite element simulations. In addition, an adaption of the model to noisy data is shown and its [...]

Key facts. Cited by 45 papers in the map (data through 7 Oct 2026). First posted on arXiv 5 Feb 2023.

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Citing papers per year: 2023: 8; 2024: 7; 2025: 16; 2026: 14.

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  1. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  2. 2023Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanicscited by 16
  3. 2023On sparse regression, Lp-regularization, and automated model discoverycited by 15
  4. 2025Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)cited by 12
  5. 2023Stress representations for tensor basis neural networks: alternative formulations to Finger-Rivlin-Ericksencited by 9
  6. 2024Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networkscited by 6
  7. 2024Automated Model Discovery for Tensional Homeostasis: Constitutive Machine Learning in Growth and Remodelingcited by 6
  8. 2025Polyconvex Physics-Augmented Neural Network Constitutive Models in Principal Stretchescited by 5
  9. 2026Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material modelscited by 5
  10. 2023Symmetry-enforcing neural networks with applications to constitutive modelingcited by 4
  11. 2025Physics Augmented Machine Learning Discovery of Composition-Dependent Constitutive Laws for 3D Printed Digital Materialscited by 4
  12. 2025A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strainscited by 4
33 more
  1. 2024Neural Networks for Generating Better Local Optima in Topology Optimizationcited by 3
  2. 2024Consistent machine learning for topology optimization with microstructure-dependent neural network material modelscited by 3
  3. 2025Input convex neural networks: universal approximation theorem and implementation for isotropic polyconvex hyperelastic energiescited by 3
  4. 2025EquiNO: A Physics-Informed Neural Operator for Multiscale Simulationscited by 3
  5. 2025Generalized invariants meet constitutive neural networks: A novel framework for hyperelastic materialscited by 3
  6. 2025Automated Constitutive Model Discovery by Pairing Sparse Regression Algorithms with Model Selection Criteriacited by 3
  7. 2024Similarity Equivariant Graph Neural Networks for Homogenization of Metamaterialscited by 2
  8. 2025Input Specific Neural Networkscited by 2
  9. 2025COMMET: orders-of-magnitude speed-up in finite element method via batch-vectorized neural constitutive updatescited by 2
  10. 2025Automating modeling in mechanics: LLMs as designers of physics-constrained neural networks for constitutive modeling of materialscited by 2
  11. 2026Hyperelastic constitutive model discovery with differentiable finite elements and structure-preserving neural networkscited by 2
  12. 2023Predicting and explaining nonlinear material response using deep Physically Guided Neural Networks with Internal Variablescited by 1
  13. 2023Exploring hyperelastic material model discovery for human brain cortex: multivariate analysis vs. artificial neural network approachescited by 1
  14. 2023Incorporating sufficient physical information into artificial neural networks: a guaranteed improvement via physics-based Rao-Blackwellizationcited by 1
  15. 2025Spectral Normalization and Voigt-Reuss net: A universal approach to microstructure-property forecasting with physical guaranteescited by 1
  16. 2026A Convex Route to Thermoelasticity: Learning Internal Energy and Dissipationcited by 1
  17. 2024A thermodynamically consistent physics-informed deep learning material model for short fiber/polymer nanocompositescited by 0
  18. 2024Equivariant graph convolutional neural networks for the representation of homogenized anisotropic microstructural mechanical responsecited by 0
  19. 2025Atrial constitutive neural networkscited by 0
  20. 2025Fully data-driven inverse hyperelasticity with hyper-network neural ODE fieldscited by 0
  21. 2025Multi-patch isogeometric neural solver for partial differential equations on computer-aided design domainscited by 0
  22. 2025Robust inverse material design with physical guarantees using the Voigt-Reuss Netcited by 0
  23. 2026Physics-constrained symbolic regression for discovering closed-form equations of multimodal water retention curves from experimental datacited by 0
  24. 2026Physics-Informed Reduced-Order Operator Learning for Hyperelasticity in Continuum Micromechanicscited by 0
  25. 2026LLM-driven design of physics-constrained constitutive models: two agents are better than onecited by 0
  26. 2026Finite Element-Based Material Learning via Automatic Differentiation: Learning constitutive neural network models from full-field deformation datacited by 0
  27. 2026CANN-EUCLID: unsupervised constitutive artificial neural network model discovery from full-field datacited by 0
  28. 2026Physics-Informed Discovery of Yield Functions in Plasticity via Convex Neural Representationscited by 0
  29. 2026Implementation of Hyperelastic Physics-Augmented Neural Networks in the Explicit Finite Element Codes Simcenter Radioss and OpenRadioss with Applications to Impact Eventscited by 0
  30. 2026Interval and fuzzy physics-augmented neural networks (iPANN and fPANN) for uncertainty quantification and propagation in constitutive modelingcited by 0
  31. 2026Benchmarking data-driven material models on the classic Treloar datasetcited by 0
  32. 2026Data-Driven Discovery of Composition-Dependent Constitutive Models for Hyperelasticity and Viscoelasticity of Digital Materialscited by 0
  33. 2026Nonlinear elliptic homogenization with the parametric Deep Ritz methodcited by 0