Pith. sign in

REVIEW 4 major objections 5 minor 60 references

Fully data-driven inverse hyperelasticity with hyper-network neural ODE fields

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A hyper-network of neural ODEs recovers a full polyconvex strain-energy field at every material point from full-field deformation data and boundary forces, with no closed-form constitutive equation assumed.

desk verdict A novel hyper-network NODE field method for heterogeneous material identification, with honest noise sensitivity, but the unquantified coupling between interpolation error and the inferred material field is a load-bearing caveat. read the letter →

arxiv 2506.08146 v2 pith:NMOPA5IL submitted 2025-06-09 cs.LG physics.comp-ph

Vahidullah Taç , Amirhossein Amiri-Hezaveh , Manuel K. Rausch , Grace N. Bechtel , Francisco Sahli Costabal , Adrian Buganza Tepole This is my paper
classification cs.LGphysics.comp-ph MSC 74B2074G7568T07
keywords inverseproblemhyperelasticityneuralordinarydifferentialequationsheterogeneousmaterialspolyconvexitydigitalimagecorrelationhyper-networkdata-drivenconstitutivemodeling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the mechanical properties of a heterogeneous material, including the full stress–strain response at every point rather than just a stiffness map, can be recovered from a measured full-field displacement or deformation gradient field $\mathbf{F}$ and boundary forces, without assuming a closed-form constitutive equation in advance. The material is represented as a spatial field of polyconvex strain-energy functions built from neural ordinary differential equations, so physical admissibility such as frame indifference, a stress-free natural state, and polyconvexity is enforced by construction rather than by penalties. A hyper-network maps material coordinates to NODE parameters, and training minimizes a loss that enforces the strong form of equilibrium and the traction boundary conditions. The authors demonstrate recovery on synthetic heterogeneous neo-Hookean and anisotropic (Gasser–Ogden–Holzapfel) examples, on noisy datasets where additional tests restore accuracy, and on a 3D-printed elastomer specimen with a digit-shaped stiff inclusion. If right, this removes the main bottleneck of classical inverse methods, the need to guess a material model before identifying parameters.

What carries the argument

The load-bearing object is the hyper-network NODE field: a neural ODE whose integration from $h(0)=0$ produces a monotone function of each polyconvex invariant, summed to form $\Psi_{\mathrm{NODE}}(\mathbf{X},\mathbf{C})$; a hyper-network $\mathrm{NN}_{\mathrm{hyper}}(\mathbf{X})$ generates the spatially varying NODE parameters, so the material model itself is a field. The machinery is completed by a Fourier-feature MLP that interpolates the discrete deformation gradient field into a smooth $\mathbf{F}^{\mathrm{int}}$ and by a multi-objective loss that penalizes the equilibrium residual $\nabla_{\mathbf{X}}\cdot\mathbf{P}^{\mathrm{pred}}$ and the traction mismatch on $\Gamma_t$, with two pre-training steps (a homogenized average-stress step and a hyper-network initialization) stabilizing the inverse problem. This combination is what lets the method discover arbitrary material fields, including anisotropy, without a preset constitutive ansatz.

What would settle it

Take a specimen with a known, independently characterized heterogeneous material (for example, a stiff inclusion in a soft matrix) and measure full-field displacement and boundary tractions under one loading mode; if the NODE material field identified from that single experiment cannot predict the measured force–displacement response under a second, different loading mode, the claim that the method recovers the true material field is falsified. A sharper version: use two materials that behave identically under uniaxial tension but differently under shear; the method would conflate them if the training data does not activate shear, so a subsequent shear test would expose the gap.

Watch

Extended reading notes

Core claim

The central discovery is that the inverse identification problem for heterogeneous hyperelasticity can be solved without parametric material families. From a continuous interpolation of the deformation gradient $\mathbf{F}^{\mathrm{int}}(\mathbf{X},t)$, obtained by Fourier-feature neural interpolation of finite-difference deformation gradients, and from measured boundary tractions, the framework learns a first Piola–Kirchhoff stress field $\mathbf{P}^{\mathrm{pred}}(\mathbf{X},t)$ that satisfies $\nabla_{\mathbf{X}}\cdot\mathbf{P}^{\mathrm{pred}}=0$ and, at every point $\mathbf{X}$, a polyconvex strain-energy function $\Psi_{\mathrm{NODE}}(\mathbf{X},I_1,I_2,I_{4v},I_{4w},J)$ expressed as a sum of neural ODEs over polyconvex invariants. Heterogeneity appears as a hyper-network output: a map from $\mathbf{X}$ to all NODE parameters. In the synthetic examples, the learned stress–strain responses reproduce the ground-truth neo-Hookean and GOH models at sampled points; the experimental example yields a material field whose stiff-inclusion/soft-matrix structure matches the printed specimen.

Load-bearing premise

The load-bearing premise is that the continuously interpolated deformation gradient field $\mathbf{F}^{\mathrm{int}}$ is accurate enough that the equilibrium residual $\nabla_{\mathbf{X}}\cdot\mathbf{P}$ computed from it reflects the true physics rather than interpolation error; if the interpolation is inaccurate, the training can absorb the error into the learned material field and report heterogeneity that is an artifact of the smoothing step.

Editorial extensions

If this is right

  • Full-field imaging plus boundary force data becomes sufficient to reconstruct a complete hyperelastic constitutive model at every material point, not merely a stiffness or modulus map.
  • Anisotropy is discovered rather than assumed: the GOH-inclusion example recovers an anisotropic response inside the inclusion and an isotropic response in the surrounding matrix without any input about fiber directions.
  • Noise tolerance is data-dependent in a practical way: a single noisy dataset with roughly 6% error in the deformation gradient degrades identification, while two or three datasets (repeated tests or additional loading modes) restore accurate material recovery.
  • Classical inverse methods that require a closed-form material law and prior knowledge of phase geometry can in principle be replaced by a single pipeline that outputs both an equilibrium stress field and a per-point material model.
  • Because the strain energy is polyconvex by construction, the recovered models inherit the well-posedness properties that follow from polyconvexity, rather than relying on ad hoc regularization to enforce physical admissibility.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's noise study suggests a practical operating rule it does not state: the quantity to monitor is the signal-to-noise ratio in $\mathbf{F}$, and around 19 dB (about 6% error) separates single-test identification from needing multi-test data.
  • The hyper-network's coordinate-to-parameter map is a smooth field, so the method as presented cannot represent sharp material interfaces perfectly; interface-aware interpolation or a discontinuous hyper-network would be a natural extension that the ablation study does not explore.
  • If the interpolated deformation gradient is the main error source, then comparing identification results obtained with displacement interpolation versus direct deformation-gradient interpolation on the same experimental data would isolate how much of the recovered material field is an artifact of the smoothing step.
  • The per-point NODE energies could be reused as priors for Bayesian or generative identification approaches; the discussion notes that probabilistic machinery over NODE material classes exists, and combining it with the present deterministic solver might give calibrated uncertainty estimates for the recovered heterogeneity.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a fully data-driven method for identifying heterogeneous hyperelastic material fields from full-field deformation measurements and boundary tractions. Displacements (or deformation gradients) are processed through a finite-difference scheme and then interpolated by a Fourier-feature neural network; the resulting continuous deformation gradient field is used in a loss that enforces the strong form of equilibrium and Neumann boundary conditions. The material model at each point is a NODE-based, polyconvex strain-energy potential whose parameters are generated by a hypernetwork. The paper demonstrates the approach on synthetic heterogeneous neo-Hookean and GOH examples, a noise study with repeated or multi-observation data, and a 3D-printed MNIST-geometry specimen tested with DIC.

Significance. If the identification is unbiased, the framework is a valuable contribution: it removes the need for a closed-form constitutive ansatz, returns a full material model field, avoids repeated forward solves, and is validated against independent JAX-FEM ground truths. The authors also provide public code and a useful ablation study of the interpolation, NODE, and hypernetwork architectures. However, the central claim of robust, unbiased constitutive identification is not yet fully supported because the paper does not quantify the coupling between interpolation error in the deformation gradient and the learned material field, and the experimental demonstration remains qualitative.

major comments (4)
  1. [§5.3, Eq. (13), Eq. (18)] The equilibrium loss is evaluated with the interpolated field F_int, whose derivatives are not constrained by the data-fitting loss in Eq. (13). Any interpolation error, especially in the gradients of F_int near material interfaces, can be absorbed by the hyper-network/NODE material field while reducing the equilibrium residual. The identified material may then satisfy equilibrium on F_int but not on the true deformation field, which directly threatens the claim that the true constitutive field is recovered. The manuscript does not report quantitative interpolation errors for F_int or its derivatives, nor a post-training equilibrium residual evaluated on the true F. Please add L2 and H1 error norms for F_int relative to ground truth, a post-training equilibrium residual on the true deformation, and a convergence or sensitivity study with respect to interpolation resolution and architecture.
  2. [§2.3, Fig. 4] The text states that "about 0.06 error is tolerated in the non-dimensional input F," but Fig. 4d shows that a single noisy observation with approximately 6% error in F fails to recognize the ring inhomogeneity. Recovery in Fig. 4e is achieved only by using repeated biaxial tests or additional loading modes. This is an important qualification of the robustness claim: 6% F error is not tolerated by a single experiment. Please restate the noise limitation precisely and quantify the improvement from multi-observation data, for example by reporting error norms of the identified material field as a function of the number of tests and loading modes.
  3. [§2.4, Fig. 5] The experimental example is only qualitative. No quantitative error is reported between the identified material field and the reference homogeneous-sample responses, the DIC displacement noise level is not estimated, and the uniaxial comparison curves have no uncertainty bands or error norms. Since this example is the main evidence that the method works outside synthetic settings, please include quantitative metrics such as relative error in the identified stress trace, estimated tangent moduli for inclusion and matrix, and residual norms after training.
  4. [Eq. (18) and Section 5.3] The loss weight λ for the Neumann boundary term is introduced as a hyperparameter, but its numerical value and sensitivity are never reported. Because the identified material field depends on the balance between equilibrium and boundary data, the absence of this value impairs reproducibility and makes it difficult to assess whether the boundary term is actually influential. Please report the value(s) of λ used and, if possible, a brief sensitivity study.
minor comments (5)
  1. [Eq. (9), Section 5.2] The notation in Eq. (9) is confusing: the initial condition is written both as h(0)=X and h(0)=dψ/dI_poly. Presumably the NODE maps an invariant to a derivative of the energy; please clarify the symbols and the dimension of the ODE.
  2. [Section 3, Discussion] The sentence "Errors in strain or deformation gradient of up to 6 perfect were handled well" contains a typo: "6 perfect" should be "6 percent," and the wording should be reconciled with the failure shown in Fig. 4d for a single noisy observation.
  3. [Fig. 8 caption and Section 9.1] The caption of Fig. 8d lists Fourier features [2,40] for the finest hypernetwork, whereas the text says [2,80]. Please correct the inconsistency.
  4. [Section 9.2] The text describing panel (d) of Fig. 9 says "the final result after the second pre-training," but the panel appears to show the main training result. Please fix this wording so the training stages are unambiguous.
  5. [Sections 5.1 and 5.2] The anisotropic invariants I_{4v} and I_{4w} and the associated structural tensors are used without definition in this paper; readers need at least the definitions or a precise pointer to the previous NODE paper to reproduce the energy representation in Eq. (10).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the identified material field is validated against independently generated ground-truth material models and separate experimental tests, and the self-cited NODE architecture is an externally published, benchmarked building block rather than a result whose validity depends on the present paper.

full rationale

The paper's central derivation is an inverse problem: given an interpolated deformation gradient field F_int (Eq. 13) and boundary traction data, optimize the hyper-network parameters so that the NODE-based stress P_pred = 2 F_int dPsi_NODE/dC_int satisfies equilibrium and Neumann boundary conditions (Eq. 18). The objective (18) contains no quantity that is defined by the ground-truth material field or by the validation outputs; its optimum is set entirely by the deformation data, the boundary forces, and the polyconvex NODE ansatz. The claimed results are then compared against ground truth generated with JAX-FEM using neo-Hookean and GOH models, which are independent of the NODE representation, and against separate homogeneous experimental samples in the MNIST example. Thus, the validation is not a renaming of the fitted output. The self-citations, most notably [36] for the NODE polyconvex constitutive representation and [54] for anisotropic NODE models, are load-bearing architectural choices, but they cite previously published, benchmarked work with independent content; the present paper does not invoke any uniqueness theorem from its own authors to forbid alternative material models. The principal limitation visible in the text is numerical: interpolation error in F_int can be absorbed into the material field because only equilibrium (not stress data) constrains the NODE parameters. That is a real identifiability and error-propagation concern, but it is not circular in the sense of a fitted parameter being renamed as a prediction or a derivation reducing to its own input by construction. No circular step can be exhibited by quoting an equation that equates a claimed output to a fitted input, so the appropriate finding is no significant circularity, score 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the expressive power of the NODE representation and the continuity of the hyper-network field, plus standard continuum mechanics assumptions. The method does not introduce new physical constants; the loss weight λ is a free hyperparameter.

free parameters (1)
  • λ (Neumann boundary loss weight) = not reported
    In Eq. (18), the traction boundary condition term is weighted by λ. The paper does not specify its value or how it was tuned, which affects the balance between equilibrium satisfaction and boundary force matching.
assumptions (5)
  • ad hoc to paper The NODE representation in Eq. (10) is expressive enough to approximate the true hyperelastic material behavior in the domain.
    The paper assumes this specific sum of NODEs over polyconvex invariants can represent arbitrary materials, including the GOH model with two fiber families. No approximation theorem is proven for this exact function class.
  • domain assumption The material property field is a continuous function of position X, representable by the chosen hyper-network architecture with Fourier features.
    Eq. (11) maps X to NODE parameters via an MLP. The method relies on this smoothness, though Fourier features are used to approximate sharp interfaces.
  • domain assumption The interpolated deformation gradient field F_int is differentiable and accurate enough for computing the equilibrium residual in Eq. (18).
    The equilibrium loss requires computing the divergence of P, which involves derivatives of F_int. Interpolation errors can bias the identified material field.
  • standard math Polyconvexity of the strain energy is a necessary and sufficient condition for physical admissibility, and the NODE construction in Eq. (9) ensures it.
    Based on Ball's theorem [57] and the construction from [36], used in Section 5.2 to build the constitutive model.
  • standard math The strong form of equilibrium (∇·P=0) and the traction boundary conditions are the correct governing equations for the quasistatic problem.
    These are the standard balance laws stated in Eq. (2) and used to define the loss function in Eq. (18).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fully data-driven inverse hyperelasticity with hyper-network neural ODE fields." pith.science (2026). https://pith.science/paper/NMOPA5IL

@misc{pith2026250608146,
  author       = {Pith},
  title        = {Pith review of: Fully data-driven inverse hyperelasticity with hyper-network neural ODE fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMOPA5IL}},
  note         = {Machine review of arXiv:2506.08146}
}
read the original abstract

We propose a new framework for identifying mechanical properties of heterogeneous materials without a closed-form constitutive equation. Given a full-field measurement of the displacement field, for instance as obtained from digital image correlation (DIC), a continuous approximation of the strain field is obtained by training a neural network that incorporates Fourier features to effectively capture sharp gradients in the data. A physics-based data-driven method built upon ordinary neural differential equations (NODEs) is employed to discover constitutive equations. The NODE framework can represent arbitrary materials while satisfying constraints in the theory of constitutive equations by default. To account for heterogeneity, a hyper-network is defined, where the input is the material coordinate system, and the output is the NODE-based constitutive equation. The parameters of the hyper-network are optimized by minimizing a multi-objective loss function that includes penalty terms for violations of the strong form of the equilibrium equations of elasticity and the associated Neumann boundary conditions. We showcase the framework with several numerical examples, including heterogeneity arising from variations in material parameters, spatial transitions from isotropy to anisotropy, material identification in the presence of noise, and, ultimately, application to experimental data. As the numerical results suggest, the proposed approach is robust and general in identifying the mechanical properties of heterogeneous materials with very few assumptions, making it a suitable alternative to classical inverse methods.

Figures

Figures reproduced from arXiv: 2506.08146 by the authors.

Figure 1
Figure 1. Detailed overview of the data-driven inverse method. The framework begins with raw displacement [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Identification of inhomogeneity through the proposed data-driven inverse method. A soft ”P”- [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Discovering material anisotropy. A GOH material elliptical inclusion is embedded in a neo-Hookean [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Effect of noise. A ring-shape inclusion is embedded within a square matrix domain, with both [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 2
Figure 2. Figure 2: a) 3D printed sample of heterogeneous material and testing b) Displacement field from DIC c) NN interpolated strains d) Ground truth 0.25 0.05 x u y u -0.04 0.04 -0.05 -0.25 0.4 0.1 (MPa) (m) (m) e) Inverse identification of material field(MPa) [PITH_FULL_IMAGE:figure…
Figure 5
Figure 5. Figure 5: Characterizing heterogeneous materials from experimental data on 3D printed elastomers. (a) [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Ablation study to demonstrate the effect of Fourier features used in the interpolation of ground [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Ablation study to show the effect of the architecture used in the core NODEs. The ground truth [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Ablation study to indicate how various architecture used in hyper-network can affect the final [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: An illustrative example showing the results of the major steps in the proposed inverse method. [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

60 extracted references · 56 canonical work pages

  1. [47]

    R. Shi, H. Yang, J. Chen, K. Hackl, S. Avril, Y. He, Deep learning without stress data on the discovery of multi-regional hyperelastic properties, Computational Mechanics (2025) 1–30

    work page 2025

  2. [1]

    Sharma, T

    A. Sharma, T. Mukhopadhyay, S. M. Rangappa, S. Siengchin, V. Kushvaha, Advances in computational intelligence of polymer composite materials: machine learning assisted modeling, analysis and design, Archives of Computational Methods in Engineering 29 (5) (2022) 3341–3385

    work page 2022

  3. [2]

    C.-Y. Lin, M. Mathur, M. Malinowski, T. A. Timek, M. K. Rausch, The impact of thickness heterogeneity on soft tissue biomechanics: a novel measurement technique and a demonstration on heart valve tissue, Biomechanics and modeling in mechanobiology 22 (5) (2023) 1487–1498

    work page 2023

  4. [3]

    Sack, Magnetic resonance elastography from fundamental soft-tissue mechanics to diagnostic imaging, Nature Reviews Physics 5 (1) (2023) 25–42

    I. Sack, Magnetic resonance elastography from fundamental soft-tissue mechanics to diagnostic imaging, Nature Reviews Physics 5 (1) (2023) 25–42

    work page 2023

  5. [4]

    Moreno-Flores, M

    O. Moreno-Flores, M. K. Rausch, A. B. Tepole, The role of interface geometry and appendages on the mesoscale mechanics of the skin, Biomechanics and Modeling in Mechanobiology 23 (2) (2024) 553–568

    work page 2024

  6. [5]

    Koorman, K

    T. Koorman, K. A. Jansen, A. Khalil, P. D. Haughton, D. Visser, M. A. R¨ atze, W. E. Haakma, G. Sakalauskait` e, P. J. van Diest, J. de Rooij, et al., Spatial collagen stiff- ening promotes collective breast cancer cell invasion by reinforcing extracellular matrix alignment, Oncogene 41 (17) (2022) 2458–2469

    work page 2022

  7. [6]

    L. G. Olson, R. D. Throne, A. J. Nolte, A. Crump, K. Griffin, T. Han, N. Iovanac, T. Janssen, M. Jones, X. Ling, et al., An inverse problem approach to stiffness mapping for early detection of breast cancer: tissue phantom experiments, Inverse Problems in Science and Engineering 27 (7) (2019) 1006–1037

    work page 2019

  8. [7]

    C.-K. Chai, A. C. Akyildiz, L. Speelman, F. J. Gijsen, C. W. Oomens, M. R. van Sambeek, A. van der Lugt, F. P. Baaijens, Local anisotropic mechanical properties of human carotid atherosclerotic plaques–characterisation by micro-indentation and inverse finite element analysis, Journal of the mechanical behavior of biomedical materials 43 (2015) 59–68

    work page 2015

Show all 60 references
  1. [8]

    Zhang, V

    Y. Zhang, V. Y. Wang, A. E. Morgan, J. Kim, R. Tafreshi, A. W. Wallace, J. M. Guccione, J. W. Weinsaft, L. Ge, M. B. Ratcliffe, Finite-element based optimization of left ventricular passive stiffness in normal volunteers and patients after myocardial infarction: utility of an ...

    2021

  2. [9]

    L. Qian, H. Zhao, Nanoindentation of soft biological materials, Micromachines 9 (12) (2018) 654

    2018

  3. [10]

    E. U. Azeloglu, K. D. Costa, Atomic force microscopy in mechanobiology: measuring microelastic heterogeneity of living cells, Atomic force microscopy in biomedical research: methods and protocols (2011) 303–329. 24

    2011

  4. [11]

    Pailler-Mattei, S

    C. Pailler-Mattei, S. Bec, H. Zahouani, In vivo measurements of the elastic mechanical properties of human skin by indentation tests, Medical engineering & physics 30 (5) (2008) 599–606

    2008

  5. [12]

    A. O. Moghaddam, J. Wei, J. Kim, A. C. Dunn, A. J. W. Johnson, An indentation-based approach to determine the elastic constants of soft anisotropic tissues, Journal of the Mechanical Behavior of Biomedical Materials 103 (2020) 103539

    2020

  6. [13]

    M. A. Cox, N. J. Driessen, C. V. Bouten, F. P. Baaijens, Mechanical characterization of anisotropic planar biological soft tissues using large indentation: a computational feasibility study, Journal of biomechanical engineering 128 (3) (2006) 428–436

    2006

  7. [14]

    D. M. Kingsley, C. H. McCleery, C. D. Johnson, M. T. Bramson, D. Rende, R. J. Gilbert, D. T. Corr, Multi-modal characterization of polymeric gels to determine the influence of testing method on observed elastic modulus, Journal of the mechanical behavior of biomedical material...

    2019

  8. [15]

    C. M. Luetkemeyer, U. Scheven, J. B. Estrada, E. M. Arruda, Constitutive modeling of the anterior cruciate ligament bundles and patellar tendon with full-field methods, Journal of the Mechanics and Physics of Solids 156 (2021) 104577

    2021

  9. [16]

    P. K. Rastogi, Photomechanics, Vol. 77, Springer Science & Business Media, 2003

    2003

  10. [17]

    Claire, F

    D. Claire, F. Hild, S. Roux, A finite element formulation to identify damage fields: the equilibrium gap method, International journal for numerical methods in engineering 61 (2) (2004) 189–208

    2004

  11. [18]

    Avril, M

    S. Avril, M. Bonnet, A.-S. Bretelle, M. Gr´ ediac, F. Hild, P. Ienny, F. Latourte, D. Lemosse, S. Pagano, E. Pagnacco, et al., Overview of identification methods of mechanical parameters based on full-field measurements, Experimental Mechanics 48 (2008) 381–402

    2008

  12. [19]

    Genovese, P

    K. Genovese, P. Badel, C. Cavinato, B. Pierrat, M. Bersi, S. Avril, J. Humphrey, Multi- view digital image correlation systems for in vitro testing of arteries from mice to humans, Experimental mechanics 61 (2021) 1455–1472

    2021

  13. [20]

    J. B. Estrada, C. M. Luetkemeyer, U. M. Scheven, E. M. Arruda, Mr-u: material char- acterization using 3d displacement-encoded magnetic resonance and the virtual fields method, Experimental Mechanics 60 (2020) 907–924

    2020

  14. [21]

    Toussaint, M

    E. Toussaint, M. Gr´ ediac, F. Pierron, The virtual fields method with piecewise virtual fields, International Journal of Mechanical Sciences 48 (3) (2006) 256–264

    2006

  15. [22]

    Gr´ ediac, F

    M. Gr´ ediac, F. Pierron, Applying the virtual fields method to the identification of elasto- plastic constitutive parameters, International Journal of Plasticity 22 (4) (2006) 602–627

    2006

  16. [23]

    Pierron, M

    F. Pierron, M. Gr´ ediac, The virtual fields method: extracting constitutive mechani- cal parameters from full-field deformation measurements, Springer Science & Business Media, 2012. 25

    2012

  17. [24]

    K. T. Kavanagh, R. W. Clough, Finite element applications in the characterization of elastic solids, International Journal of Solids and Structures 7 (1) (1971) 11–23

    1971

  18. [25]

    Pottier, F

    T. Pottier, F. Toussaint, P. Vacher, Contribution of heterogeneous strain field measure- ments and boundary conditions modelling in inverse identification of material parame- ters, European Journal of Mechanics-A/Solids 30 (3) (2011) 373–382

    2011

  19. [26]

    Geymonat, F

    G. Geymonat, F. Hild, S. Pagano, Identification of elastic parameters by displacement field measurement, Comptes Rendus Mecanique 330 (6) (2002) 403–408

    2002

  20. [27]

    Florentin, G

    E. Florentin, G. Lubineau, Identification of the parameters of an elastic material model using the constitutive equation gap method, Computational Mechanics 46 (2010) 521– 531

    2010

  21. [28]

    Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, SIAM Journal on Applied Mathematics 50 (6) (1990) 1635–1644

    M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, SIAM Journal on Applied Mathematics 50 (6) (1990) 1635–1644

    1990

  22. [29]

    Claire, F

    D. Claire, F. Hild, S. Roux, Identification of a damage law by using full-field displacement measurements, International Journal of Damage Mechanics 16 (2) (2007) 179–197

    2007

  23. [30]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics- informed machine learning, Nature Reviews Physics 3 (6) (2021) 422–440

    2021

  24. [31]

    Ghaboussi, J

    J. Ghaboussi, J. Garrett Jr, X. Wu, Knowledge-based modeling of material behavior with neural networks, Journal of engineering mechanics 117 (1) (1991) 132–153

    1991

  25. [32]

    As’ ad, P

    F. As’ ad, P. Avery, C. Farhat, A mechanics-informed artificial neural network approach in data-driven constitutive modeling, International Journal for Numerical Methods in Engineering 123 (12) (2022) 2738–2759

    2022

  26. [33]

    J. N. Fuhg, N. Bouklas, On physics-informed data-driven isotropic and anisotropic con- stitutive models through probabilistic machine learning and space-filling sampling, Com- puter Methods in Applied Mechanics and Engineering 394 (2022) 114915

    2022

  27. [34]

    Linden, D

    L. Linden, D. K. Klein, K. A. Kalina, J. Brummund, O. Weeger, M. K¨ astner, Neural networks meet hyperelasticity: A guide to enforcing physics, Journal of the Mechanics and Physics of Solids 179 (2023) 105363

    2023

  28. [35]

    Ta¸ c, K

    V. Ta¸ c, K. Linka, F. Sahli-Costabal, E. Kuhl, A. B. Tepole, Benchmarking physics- informed frameworks for data-driven hyperelasticity, Computational Mechanics (Jun. 2023).doi:10.1007/s00466-023-02355-2

    2023 doi

  29. [36]

    V. Tac, F. Sahli Costabal, A. B. Tepole, Data-driven tissue mechanics with polyconvex neural ordinary differential equations, Computer Methods in Applied Mechanics and Engineering 398 (2022) 115248.doi:https://doi.org/10.1016/j.cma.2022.115248

    2022

  30. [37]

    Kirchhoff, D

    J. Kirchhoff, D. Luo, T. O’Leary-Roseberry, O. Ghattas, Inference of heterogeneous ma- terial properties via infinite-dimensional integrated dic, arXiv preprint arXiv:2408.10217 (2024). 26

    2024 arXiv

  31. [38]

    T. Xue, S. Liao, Z. Gan, C. Park, X. Xie, W. K. Liu, J. Cao, JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science

  32. [39]

    T. C. Gasser, R. W. Ogden, G. A. Holzapfel, Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, Journal of the royal society interface 3 (6) (2005) 15–35

    2005

  33. [40]

    Grediac, F

    M. Grediac, F. Pierron, S. Avril, E. Toussaint, The virtual fields method for extracting constitutive parameters from full-field measurements: a review, Strain 42 (4) (2006) 233–253

    2006

  34. [41]

    M. A. Sutton, J. Yan, S. Avril, F. Pierron, S. M. Adeeb, Identification of heterogeneous constitutive parameters in a welded specimen: uniform stress and virtual fields methods for material property estimation, Experimental Mechanics 48 (2008) 451–464

    2008

  35. [42]

    Ereiz, I

    S. Ereiz, I. Duvnjak, J. F. Jim´ enez-Alonso, Review of finite element model updating methods for structural applications, in: Structures, Vol. 41, Elsevier, 2022, pp. 684–723

    2022

  36. [43]

    Vald´ es-Alonzo, C

    G. Vald´ es-Alonzo, C. Binetruy, B. Eck, A. Garc ´ ıa-Gonz´ alez, A. Leygue, Phase distribu- tion and properties identification of heterogeneous materials: A data-driven approach, Computer Methods in Applied Mechanics and Engineering 390 (2022) 114354

    2022

  37. [44]

    W. Wu, M. Daneker, K. T. Turner, M. A. Jolley, L. Lu, Identifying heterogeneous micromechanical properties of biological tissues via physics-informed neural networks, Small Methods (2025) 2400620

    2025

  38. [45]

    Y. Liu, Y. Chen, B. Ding, Deep learning in frequency domain for inverse identification of nonhomogeneous material properties, Journal of the Mechanics and Physics of Solids 168 (2022) 105043

    2022

  39. [46]

    C.-T. Chen, G. X. Gu, Learning hidden elasticity with deep neural networks, Proceedings of the National Academy of Sciences 118 (31) (2021) e2102721118

    2021

  40. [48]

    G. A. Padmanabha, J. N. Fuhg, C. Safta, R. E. Jones, N. Bouklas, Improving the performance of stein variational inference through extreme sparsification of physically- constrained neural network models, Computer Methods in Applied Mechanics and En- gineering 432 (2024) 117359

    2024

  41. [49]

    D. V. Patel, D. Ray, A. A. Oberai, Solution of physics-based bayesian inverse problems with deep generative priors, Computer Methods in Applied Mechanics and Engineering 400 (2022) 115428

    2022

  42. [50]

    Lejeune, Mechanical mnist: A benchmark dataset for mechanical metamodels, Ex- treme Mechanics Letters 36 (2020) 100659

    E. Lejeune, Mechanical mnist: A benchmark dataset for mechanical metamodels, Ex- treme Mechanics Letters 36 (2020) 100659. 27

    2020

  43. [51]

    V. Tac, M. K. Rausch, I. Bilionis, F. Sahli Costabal, A. B. Tepole, Generative hyperelas- ticity with physics-informed probabilistic diffusion fields, Engineering with Computers 41 (1) (2025) 51–69

    2025

  44. [52]

    Siebert, T

    T. Siebert, T. Becker, K. Spiltthof, I. Neumann, R. Krupka, Error estimations in digital image correlation technique, Applied mechanics and materials 7 (2007) 265–270

    2007

  45. [53]

    Bornert, F

    M. Bornert, F. Br´ emand, P. Doumalin, J.-C. Dupr´ e, M. Fazzini, M. Gr´ ediac, F. Hild, S. Mistou, J. Molimard, J.-J. Orteu, et al., Assessment of digital image correlation measurement errors: methodology and results, Experimental mechanics 49 (2009) 353– 370

    2009

  46. [54]

    Ta¸ c, M

    V. Ta¸ c, M. K. Rausch, F. S. Costabal, B. Tepole, Data-driven anisotropic finite vis- coelasticity using neural ordinary differential equations, Computer Methods in Applied Mechanics and Engineering 411 (2023)

    2023

  47. [55]

    Itskov, et al., Tensor algebra and tensor analysis for engineers, Springer, 2007

    M. Itskov, et al., Tensor algebra and tensor analysis for engineers, Springer, 2007

    2007

  48. [56]

    J. M. Zhang, J. Rychlewski, Structural tensors for anisotropic solids, Archives of Me- chanics 42 (3) (1990) 267–277

    1990

  49. [57]

    J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Archive for rational mechanics and Analysis 63 (4) (1976) 337–403

    1976

  50. [58]

    Hartmann, P

    S. Hartmann, P. Neff, Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility, International journal of solids and structures 40 (11) (2003) 2767–2791

    2003

  51. [59]

    Schr¨ oder, P

    J. Schr¨ oder, P. Neff, Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions, International journal of solids and structures 40 (2) (2003) 401–445

    2003

  52. [60]

    R. T. Chen, Y. Rubanova, J. Bettencourt, D. K. Duvenaud, Neural ordinary differential equations, Advances in neural information processing systems 31 (2018). 28

    2018

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.