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FE-MAD places constitutive neural networks inside a JAX finite element solver and identifies their parameters by gradient descent on the mismatch between predicted and measured full-field deformations.

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2026-06-30 14:54 UTC pith:BJFHOQPY

load-bearing objection FE-MAD embeds a constitutive NN inside a JAX-FEM solver and backprops through the Newton loop to fit parameters directly to full-field DIC data.

arxiv 2606.05199 v1 pith:BJFHOQPY submitted 2026-05-22 physics.comp-ph cs.AI

Finite Element-Based Material Learning via Automatic Differentiation: Learning constitutive neural network models from full-field deformation data

Matthias Knipper , Chenyi Ji , Malte Brand , Kevin Linka This is my paper
classification physics.comp-ph cs.AI
keywords constitutive neural networksfinite element methodautomatic differentiationmaterial parameter identificationhyperelasticityfull-field datadigital image correlationJAX
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that an end-to-end differentiable finite element pipeline can optimize the internal parameters of a constitutive neural network so that simulated displacements and forces match experimental observations across heterogeneous tests. A sympathetic reader would care because this removes the need for separate homogeneous calibration experiments, manual derivative derivations, or large external training sets that other data-driven approaches require. The method is shown for both a flexible grey-box polyconvex network and an interpretable white-box expert network on three open datasets covering full DIC fields, reduced one-dimensional data plus global force, and simultaneous identification of matrix and inclusion phases that then generalize to new samples.

Core claim

FE-MAD integrates a constitutive neural network model within a JAX-FEM nonlinear solver and identifies its parameters through gradient-based minimization of a measurement-mismatch loss, with Newton tangent stiffness and loss gradients computed automatically using forward- and reverse-mode automatic differentiation throughout the pipeline. The approach is demonstrated for incompressible isotropic hyperelasticity using a grey-box polyconvex fully-connected CANN and a white-box expert-system CANN on three experimental datasets: perforated tensile specimens with DIC, a reduced-data stretch-plus-force case, and a heterogeneous matrix-inclusion system where both phases are recovered and tested on

What carries the argument

The end-to-end differentiable JAX-FEM pipeline that couples the constitutive neural network directly to the solver, allowing automatic computation of both the tangent stiffness matrix for Newton iterations and the gradients of the measurement-mismatch loss with respect to network parameters.

Load-bearing premise

The selected neural network architectures are expressive enough and the full-field data informative enough to recover unique, physically valid parameters that do not overfit noise or converge to non-physical local minima.

What would settle it

If the parameters recovered by FE-MAD produce large discrepancies when the learned model is applied to an independent experiment with new geometry or loading conditions not present in the identification data, the claim of reliable generalizable identification would be falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Material parameters for both high-flexibility grey-box and interpretable white-box constitutive networks can be obtained directly from heterogeneous full-field measurements.
  • No analytic adjoints or offline surrogate models are required because all necessary derivatives are obtained automatically.
  • The same framework recovers parameters in full DIC, reduced one-dimensional plus global force, and simultaneous multi-phase identification settings.
  • Learned models generalize to previously unseen samples in the matrix-inclusion demonstration.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be extended to path-dependent materials by incorporating history variables into the same differentiable solver loop.
  • Joint optimization of specimen geometry and material parameters becomes feasible within one gradient-based loop.
  • The approach may allow experimenters to design more informative specimen shapes that maximize information gain per test rather than relying on standard homogeneous geometries.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper introduces FE-MAD, an end-to-end differentiable framework that integrates constitutive artificial neural network (CANN) models into a JAX-based finite element nonlinear solver. Parameters are identified via gradient-based minimization of a measurement-mismatch loss, with Newton tangent stiffness and loss gradients obtained automatically through forward- and reverse-mode automatic differentiation. The method is demonstrated for grey-box (polyconvex fully-connected) and white-box (expert-system) CANN architectures on three open experimental datasets for incompressible isotropic hyperelasticity, including full DIC data from a perforated tensile specimen, a reduced-data one-dimensional stretch plus force-displacement case, and a heterogeneous matrix-inclusion system with generalization to 22 unseen samples.

Significance. If the results hold, FE-MAD offers a computationally tractable route to calibrating high-dimensional constitutive models directly from heterogeneous full-field data without analytic adjoints or offline surrogates. The end-to-end use of automatic differentiation in the solver pipeline is a clear technical strength, and the evaluation across multiple datasets with an explicit generalization test provides concrete evidence of practical utility. The dual grey-box/white-box architecture choice also allows both flexibility and interpretability to be assessed within the same framework.

minor comments (3)
  1. The precise form of the data-mismatch loss, any regularization terms, and the treatment of incompressibility constraints (e.g., via penalty, Lagrange multipliers, or mixed formulation) are not fully detailed in the provided abstract; explicit equations or pseudocode in §3 or §4 would improve reproducibility.
  2. The manuscript should clarify the number of trainable parameters and the specific network widths/depths for both CANN architectures, as this directly affects the claim of handling high-dimensional parameter spaces.
  3. Figure captions and the description of the matrix-inclusion generalization test would benefit from explicit mention of the error metrics used to quantify performance on the 22 unseen samples.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of FE-MAD, the recognition of its technical strengths in end-to-end automatic differentiation, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents an end-to-end differentiable finite-element framework (FE-MAD) that embeds a constitutive neural network inside a JAX-based nonlinear solver and identifies its parameters by gradient-based minimization of a measurement-mismatch loss against external experimental DIC data. The derivation chain consists of standard forward/reverse-mode AD steps for tangent stiffness and loss gradients; no equation reduces a fitted quantity to a prediction of the same quantity by construction, no uniqueness theorem is imported from the authors' prior work to force the architecture, and no ansatz is smuggled via self-citation. The central claim therefore remains self-contained against the supplied full-field measurements and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on standard continuum-mechanics assumptions for hyperelasticity plus the modeling choice that the selected NN families can represent the true constitutive response; no new physical entities are introduced.

free parameters (1)
  • CANN network weights
    Learned by gradient descent on the measurement-mismatch loss; their values are the output of the method rather than fixed inputs.
axioms (2)
  • domain assumption The materials under study are incompressible and isotropic hyperelastic
    Explicitly stated as the modeling focus; required for the strain-energy formulations used.
  • standard math The finite-element discretization and Newton solver converge to the correct weak solution for the chosen constitutive model
    Implicit in any FEM pipeline; not re-derived in the abstract.

pith-pipeline@v0.9.1-grok · 5822 in / 1411 out tokens · 35286 ms · 2026-06-30T14:54:06.831468+00:00 · methodology

0 comments
read the original abstract

The identification of constitutive neural network models from heterogeneous full-field deformation data provides a robust alternative to traditional calibration methods based on homogeneous stress-strain experiments, particularly given the high dimensionality of trainable parameters. Existing approaches must balance generality, robustness, and computational efficiency: Conventional finite element model updating is broadly applicable but computationally demanding; weak-form methods offer efficiency but are sensitive to noise and data scarcity; neural operator models are highly expressive but require extensive training datasets. This work presents FE-MAD (Finite Element-Based Material learning via Automatic Differentiation), an end-to-end differentiable framework that integrates a constitutive neural network model within a JAX-FEM nonlinear solver and identifies its parameters through gradient-based minimization of a measurement-mismatch loss. Newton tangent stiffness and loss gradients are computed automatically using forward- and reverse-mode automatic differentiation throughout the entire pipeline, thereby removing the need for analytic adjoints or offline surrogate models. FE-MAD is demonstrated for two architectures: a grey-box Constitutive Artificial Neural Network (CANN), a polyconvex, fully connected model with high flexibility, and a white-box CANN, an expert-system network with phenomenologically interpretable strain-energy terms. Focusing on incompressible isotropic hyperelasticity, FE-MAD is evaluated on three open experimental datasets: (1) full digital image correlation (DIC) of a perforated tensile specimen, (2) a reduced-data scenario with a one-dimensional stretch profile and global force-displacement curve, and (3) a heterogeneous matrix-inclusion system in which both phases constitutive laws are identified and generalized to twenty-two previously unseen samples.

Figures

Figures reproduced from arXiv: 2606.05199 by Chenyi Ji, Kevin Linka, Malte Brand, Matthias Knipper.

Figure 1
Figure 1. Figure 1: Constitutive artificial neural network architectures employed in this work. (a) Grey-box CANN: a fully connected feedforward network with arbitrary depth and width that yields a polyconvex free-energy function by construction. (b) White-box CANN: a partially connected, two-layer expert system whose eight terms recover classical phenomenological constitutive blocks (neo-Hookean, Demiray, Gent type) and ther… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the proposed end-to-end FE-MAD training loop. Full-field displacement measurements uref j obtained, e.g., from digital image correlation, together with the global reaction-force measurements F obs j , define the experimental observables. For each candidate parameter vector θ of the constitutive neural network, the JAX-FEM solver assembles the nonlinear residual, performs a Newton solve with au… view at source ↗
Figure 3
Figure 3. Figure 3: Case 1, perforated tensile specimens TT1 and TT2. Top: undeformed wide dogbone-shaped specimens with the region of interest (ROI) used for the FE model highlighted. Only the ROI is meshed with finite elements. Bottom: the same ROIs at maximum deformation, color-coded by the maximum principal stretch λ1 obtained from the FE simulation. Pronounced stretch concentrations are visible at the hole boundaries, in… view at source ↗
Figure 4
Figure 4. Figure 4: Case 1, loss evolution during training. Evolution of the displacement loss Lu (red) and the reaction-force loss LF (orange) as a function of the training epoch for the three training configurations (columns: TT1, UT & PS, UT) and the two architectures (rows: grey-box CANN, white-box CANN). The total loss minimized at each parameter update is the sum of both contributions. The high-frequency oscillations re… view at source ↗
Figure 5
Figure 5. Figure 5: Case 1, training–testing matrix of stress–stretch curves. Stress–stretch responses of all six trained models (grey￾box and white-box CANN, three training configurations) on the homogeneous tests UT and PS and on the reaction-force curve of the heterogeneous specimen TT1. Each row corresponds to a training configuration; the training data for each row is explicitly labeled. The grey-box and the white-box CA… view at source ↗
Figure 6
Figure 6. Figure 6: Case 1, displacement field on TT1. For each training configuration (columns: UT, UT & PS, TT1), the figure compares the DIC-measured vertical displacement field uy (top row, black contour lines on filled colormap) with the FE prediction (overlaid blue contours for the white-box CANN and yellow contours for the grey-box CANN), and reports the pointwise relative error of the grey-box prediction (middle row) … view at source ↗
Figure 7
Figure 7. Figure 7: Case 1, maximum principal stretch field on TT1. For each training configuration (columns: UT, UT & PS, TT1), the figure compares the DIC-derived maximum principal stretch λ1 (top row, black contour lines on filled colormap) with the FE prediction (overlaid blue contours for the white-box CANN and yellow contours for the grey-box CANN), and reports the pointwise relative error of the grey-box prediction (mi… view at source ↗
Figure 8
Figure 8. Figure 8: Case 1, distributions of the pointwise relative error on TT1. Empirical distributions of the relative error of the predicted displacement field (left) and of the maximum principal stretch (right) on the training specimen TT1, derived from Figures 6 and 7. Each violin shows the central 90 % of the data (clipped between the 5 % and 95 % percentile); the black bar indicates the central 50 % (clipped between t… view at source ↗
Figure 9
Figure 9. Figure 9: Case 1, reaction force on the unseen specimen TT2. Comparison of the measured force–displacement curve of TT2 (black) with the responses simulated by all six trained models (grey-box and white-box CANN, three training configurations). The models trained on TT1 overestimate the reaction force by approximately 9 %, whereas the models trained on UT & PS and on UT alone match the measurement to within a few pe… view at source ↗
Figure 10
Figure 10. Figure 10: Case 1, displacement field on the unseen specimen TT2. For each training configuration (columns: UT, UT & PS, TT1) and each architecture (grey-box and white-box CANN), the figure compares the DIC-measured vertical displacement field uy (top row, black contour lines on filled colormap) with the FE prediction (overlaid blue contours for the white-box CANN and yellow contours for the grey-box CANN), and repo… view at source ↗
Figure 11
Figure 11. Figure 11: Case 1, kinematic content of the training and testing data. Left: projection of the training (and testing) sets UT, PS, and TT1 into the (I1, I2) invariant plane. Center: projection of the unseen testing set TT2 into the same plane. Right: histogram of the maximum principal stretches sampled in the TT1 training set, overlaid with the relative error on UT of the models trained on TT1. Stretches above λ ≈ 3… view at source ↗
Figure 12
Figure 12. Figure 12: Case 2, five-hole tensile specimen TT. Specimen TT in the undeformed reference state (left) and at maximum deformation of uy,top = 57.3 mm (right), color-coded by the maximum principal stretch λ1. The narrow ligament between holes C1 and C3 is cut, resulting in a highly nonlinear deformation profile. The red curve indicates the path along which the maximum stretch is measured experimentally; in the undefo… view at source ↗
Figure 13
Figure 13. Figure 13: Case 2, loss evolution during training. Evolution of the stretch loss Lλ (red) and the reaction-force loss LF (orange) for the three training configurations (columns: TT, UT & PS, UT) and the two architectures (rows: grey-box CANN, white-box CANN). In contrast to Case 1, both loss contributions are now clearly visible during training because the stretch loss is markedly more sensitive to the constitutive … view at source ↗
Figure 14
Figure 14. Figure 14: Case 2, model discovery with the white-box CANN. Stress–stretch responses of the three white-box CANN models on the homogeneous tests UT and PS, and reaction-force response on the heterogeneous specimen TT. Each row corresponds to a training configuration (TT, UT & PS, UT); the training data for each row is explicitly labeled. Stacked colored areas underneath the response curves decompose the predicted fi… view at source ↗
Figure 15
Figure 15. Figure 15: Case 2, maximum principal stretch along the measurement path of TT. Comparison of the experimentally measured maximum principal stretch λ1 along the path through holes C1–C2 (black markers) with the FE predictions of the grey￾box CANN (yellow) and the white-box CANN (blue), for the three training configurations (TT, UT & PS, UT). Models trained on TT and on UT & PS recover the central peak well; models tr… view at source ↗
Figure 16
Figure 16. Figure 16: Case 2, full material response of the identified models. Stress–stretch responses of all six trained models on a broad battery of homogeneous deformations: uniaxial tension and compression (UT, UC), plane-strain tension and compression (PS, PSC), and equibiaxial bulge test (BL). The models trained on UT & PS reproduce UD and PSD most accurately and remain closest to the experimental bulge curve; the model… view at source ↗
Figure 17
Figure 17. Figure 17: Case 3, ground-truth fields of the heterogeneous material system. (a) DIC labeling of a representative thin plate specimen, with blue indicating the matrix and orange the inclusion [72]. (b) Corresponding measured vertical displacement field uy at the final load step. (c) Corresponding FE-derived maximum principal stretch field λ1 at the final load step, used as the structural ground truth against which t… view at source ↗
Figure 18
Figure 18. Figure 18: Case 3, identified heterogeneous material system applied to training and unseen specimens. Rows: training specimen (top), best-case unseen specimen (middle), worst-case unseen specimen (bottom). Columns: force–displacement curve (left), vertical displacement field uy (center), maximum principal stretch field λ1 (right). Black lines and contours indicate the experimental ground truth, orange lines and cont… view at source ↗
Figure 19
Figure 19. Figure 19: Case 3, distribution of testing losses across unseen specimens. Empirical distributions of the displacement loss Lu (left) and the reaction-force loss LF (right) across the twenty-two unseen specimens, for the grey-box and white-box CANN trained simultaneously on the matrix and on the inclusion. Red markers denote the training losses on the single training specimen; white dots indicate the median of the t… view at source ↗

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Works this paper leans on

92 extracted references · 92 canonical work pages · 3 internal anchors

  1. [1]

    H. Dal, K. Açıkgöz, and Y. Badienia. On the performance of isotropic hyperelastic constitutive models for rubber-like materials: A state of the art review.Applied Mechanics Reviews, 73(2):020802, 2021

    work page 2021

  2. [2]

    J. Tan, G. Anantha Padmanabha, S. J. Yang, and N. Bouklas. Towards rapid constitutive model discovery from multi-modal data: Physics augmented finite element model updating (paFEMU).arXiv preprint arXiv:2604.07746, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  3. [3]

    J. N. Fuhg, G. Anantha Padmanabha, N. Bouklas, B. Bahmani, W. Sun, N. N. Vlassis, M. Flaschel, P. Car- rara, and L. De Lorenzis. A review on data-driven constitutive laws for solids.Archives of Computational Methods in Engineering, 2024

    work page 2024

  4. [4]

    Regazzoni

    F. Regazzoni. The internal law of a material can be discovered from its boundary.arXiv preprint, 2026

    work page 2026

  5. [5]

    Roux and F

    S. Roux and F. Hild. Optimal procedure for the identification of constitutive parameters from experimen- tally measured displacement fields.International Journal of Solids and Structures, 184:14–23, 2020

    work page 2020

  6. [6]

    Avril, M

    S. Avril, M. Bonnet, A.-S. Bretelle, M. Grédiac, F. Hild, P. Ienny, F. Latourte, D. Lemosse, S. Pagano, E. Pagnacco, and F. Pierron. Overview of identification methods of mechanical parameters based on full-field measurements.Experimental Mechanics, 48:381–402, 2008

    work page 2008

  7. [7]

    Pierron and M

    F. Pierron and M. Grédiac. Towards material testing 2.0. a review of test design for identification of constitutive parameters from full-field measurements.Strain, 2020

    work page 2020

  8. [8]

    Römer, S

    U. Römer, S. Hartmann, J. A. Tröger, D. Anton, H. Wessels, M. Flaschel, and L. De Lorenzis. Reduced and all-at-once approaches for model calibration and discovery in computational solid mechanics.Applied Mechanics Reviews, 77:040801, 2025

    work page 2025

  9. [9]

    J. D. Collins, G. C. Hart, T. K. Hasselman, and B. Kennedy. Statistical identification of structures.AIAA Journal, 12:185–190, 1974

    work page 1974

  10. [10]

    Wiesheier, M

    S. Wiesheier, M. A. Moreno-Mateos, and P. Steinmann. Versatile data-adaptive hyperelastic energy func- tions for soft materials.Computer Methods in Applied Mechanics and Engineering, 430:117208, 2024

    work page 2024

  11. [11]

    Wiesheier, M

    S. Wiesheier, M. A. Moreno-Mateos, and P. Steinmann. Data-adaptive spline-based viscoelasticity for soft solids.Computer Methods in Applied Mechanics and Engineering, 451:118705, 2026

    work page 2026

  12. [12]

    Anton and H

    D. Anton and H. Wessels. Physics-informed neural networks for material model calibration from full-field displacement data.arXiv preprint arXiv:2212.07723, 2022

    work page arXiv 2022

  13. [13]

    Anton, Tröger J

    D. Anton, Tröger J. A., H. Wessels, U. Henkes. A Römer, and S. Hartmann. Physics-informed neural net- works for material model calibration from full-field displacement data.Advanced Modeling and Simulation in Engineering Sciences, 12(12), 2025

    work page 2025

  14. [14]

    M. Grédiac. Principle of virtual work and identification.Comptes Rendus de l’Académie des Sciences, Série II, pages 1–5, 1989

    work page 1989

  15. [15]

    Grédiac and A

    M. Grédiac and A. Vautrin. A new method for determination of bending rigidities of thin anisotropic plates.Journal of Applied Mechanics, 57:964–968, 1990. 37

    work page 1990

  16. [16]

    Grédiac, F

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

    work page 2006

  17. [17]

    Claire, F

    D. Claire, F. Hild, and S. Roux. A finite element formulation to identify damage fields: the equilibrium gap method.International Journal for Numerical Methods in Engineering, 61:189–208, 2004

    work page 2004

  18. [18]

    Flaschel, S

    M. Flaschel, S. Kumar, and L. De Lorenzis. Unsupervised discovery of interpretable hyperelastic constitu- tive laws.Computer Methods in Applied Mechanics and Engineering, 381:113852, 2021

    work page 2021

  19. [19]

    Flaschel, S

    M. Flaschel, S. Kumar, and L. De Lorenzis. Discovering plasticity models without stress data.npj Com- putational Materials, 8:91, 2022

    work page 2022

  20. [20]

    Flaschel, S

    M. Flaschel, S. Kumar, and L. De Lorenzis. Automated discovery of generalized standard material models with euclid.Computer Methods in Applied Mechanics and Engineering, 405:115867, 2023

    work page 2023

  21. [21]

    Thakolkaran, A

    P. Thakolkaran, A. Joshi, Y. Zheng, M. Flaschel, L. De Lorenzis, and S. Kumar. NN-EUCLID: Deep- learning hyperelasticity without stress data.Journal of the Mechanics and Physics of Solids, 169:105076, 2022

    work page 2022

  22. [22]

    Joshi, P

    A. Joshi, P. Thakolkaran, Y. Zheng, M. Escande, M. Flaschel, L. De Lorenzis, and S. Kumar. Bayesian- EUCLID: Discovering hyperelastic material laws with uncertainties.Computer Methods in Applied Me- chanics and Engineering, 398:115225, 2022

    work page 2022

  23. [23]

    Marino, M

    E. Marino, M. Flaschel, S. Kumar, and L. De Lorenzis. Automated identification of linear viscoelastic constitutive laws with EUCLID.Mechanics of Materials, 181:104643, 2023

    work page 2023

  24. [24]

    H. Xu, M. Flaschel, and L. De Lorenzis. Discovering non-associated pressure-sensitive plasticity models with EUCLID.Advanced Modeling and Simulation in Engineering Sciences, 12(1), 2025

    work page 2025

  25. [25]

    Ghouli, M

    S. Ghouli, M. Flaschel, S. Kumar, and L. De Lorenzis. A topology optimisation framework to design test specimens for one-shot identification or discovery of material models.Journal of the Mechanics and Physics of Solids, 203:106210, 2025

    work page 2025

  26. [26]

    Abbasi, M

    A. Abbasi, M. Ricci, P. Carrara, M. Flaschel, S. Kumar, S. Marfia, and L. De Lorenzis. Discovery of hyperelastic constitutive laws from experimental data with EUCLID.Experimental Mechanics, 2026

    work page 2026

  27. [27]

    Knauf Narouie, J.-H

    V. Knauf Narouie, J.-H. Urrea-Quintero, F. Cirak, and H. Wessels. Unsupervised constitutive model discovery from sparse and noisy data.Comput. Methods Appl. Mech. Engrg., 452(118722), 2026

    work page 2026

  28. [28]

    Ghaboussi, J

    J. Ghaboussi, J. H. Garrett, and X. Wu. Knowledge-based modeling of material behavior with neural networks.Journal of Engineering Mechanics, 117:132–153, 1991

    work page 1991

  29. [29]

    Abdolazizi, Roland C

    Kevin Linka, Markus Hillgärtner, Kian P. Abdolazizi, Roland C. Aydin, Mikhail Itskov, and Christian J. Cyron. Constitutive artificial neural networks: A fast and general approach to predictive data-driven constitutive modeling by deep learning.Journal of Computational Physics, 429:110010, 2021

    work page 2021

  30. [30]

    As’ad and C

    F. As’ad and C. Farhat. A mechanics-informed neural network framework for data-driven nonlinear vis- coelasticity, 2022

    work page 2022

  31. [31]

    D. K. Klein, M. Fernández, R. J. Martin, P. Neff, and O. Weeger. Polyconvex anisotropic hyperelasticity with neural networks.Journal of the Mechanics and Physics of Solids, 159:104703, 2022

    work page 2022

  32. [32]

    Linden, D

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

    work page 2023

  33. [33]

    K. A. Kalina, J. Brummund, W. Sun, and M. Kästner. Neural networks meet anisotropic hyperelasticity: A framework based on generalized structure tensors and isotropic tensor functions.Computer Methods in Applied Mechanics and Engineering, 437:117725, 2025

    work page 2025

  34. [34]

    D. Z. Huang, K. Xu, C. Farhat, and E. Darve. Learning constitutive relations from indirect observations using deep neural networks.Journal of Computational Physics, 416:109491, 2020. 38

    work page 2020

  35. [35]

    Tac, F.S

    V. Tac, F.S. Costabal, and A.B. Tepole. Data-driven tissue mechanics with polyconvex neural ordinary differential equations.Computer Methods in Applied Mechanics and Engineering, 398(115248), 2022

    work page 2022

  36. [36]

    Kirchdoerfer and M

    T. Kirchdoerfer and M. Ortiz. Data-driven computational mechanics.Computer Methods in Applied Mechanics and Engineering, 304:81–101, 2016

    work page 2016

  37. [37]

    Ibañez, D

    R. Ibañez, D. Borzacchiello, J. V. Aguado, E. Abisset-Chavanne, E. Cueto, P. Ladeveze, and F. Chinesta. Data-driven non-linear elasticity: constitutive manifold construction and problem discretization.Compu- tational Mechanics, 60:813–826, 2017

    work page 2017

  38. [38]

    Schoenauer, M

    M. Schoenauer, M. Sebag, F. Jouve, B. Lamy, and H. Maitournam. Evolutionary identification of macrome- chanical models. InAdvances in Genetic Programming II, pages 467–488. 1996

    work page 1996

  39. [39]

    Abdusalamov, M

    R. Abdusalamov, M. Hillg"artner, and M. Itskov. Automatic generation of interpretable hyperelastic material models by symbolic regression.International Journal for Numerical Methods in Engineering, 2023

    work page 2023

  40. [40]

    Linka and E

    K. Linka and E. Kuhl. A new family of constitutive artificial neural networks towards automated model discovery.Computer Methods in Applied Mechanics and Engineering, 403:115731, 2023

    work page 2023

  41. [41]

    Linka and E

    K. Linka and E. Kuhl. Best-in-class modeling: A novel strategy to discover constitutive models for soft matter systems.Extreme Mechanics Letters, 70:102181, 2024

    work page 2024

  42. [42]

    The language of hyperelastic materials, 2024

    Georgios Kissas, Siddhartha Mishra, Eleni Chatzi, and Laura De Lorenzis. The language of hyperelastic materials, 2024

    work page 2024

  43. [43]

    D. K. Klein, F. J. Roth, I. Valizadeh, and O. Weeger. Parametrized polyconvex hyperelasticity with physics-augmented neural networks.Data-Centric Engineering, 4:e25, 2023

    work page 2023

  44. [44]

    Chen and J

    P. Chen and J. Guilleminot. Polyconvex neural networks for hyperelastic constitutive models: A rectifica- tion approach.Mechanics Research Communications, 125:103993, 2022

    work page 2022

  45. [45]

    G. L. Geuken, P. Kurzeja, D. Wiedemann, and J. Mosler. Input convex neural networks: universal ap- proximation theorem and implementation for isotropic polyconvex hyperelastic energies.arXiv preprint arXiv:2502.08534, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  46. [46]

    Flaschel, P

    M. Flaschel, P. Steinmann, L. De Lorenzis, and E. Kuhl. Convex neural networks learn generalized standard material models.Journal of the Mechanics and Physics of Solids, 200:106103, 2025

    work page 2025

  47. [47]

    Flaschel, D

    M. Flaschel, D. Martonová, C. Veil, and E. Kuhl. Material fingerprinting: A shortcut to material model dis- covery without solving optimization problems.Computer Methods in Applied Mechanics and Engineering, 450:118573, 2026

    work page 2026

  48. [48]

    T. Xue, S. Liao, Z. Gan, and J. Cao. Jax-fem: A differentiable gpu-accelerated 3d finite element solver for automatic inverse design and mechanistic data science.Computer Physics Communications, 291:108802, 2023

    work page 2023

  49. [49]

    Van Keulen, R

    F. Van Keulen, R. Haftka, and N.-H. Kim. Review of options for structural design sensitivity analysis. part 1: linear systems.Computer Methods in Applied Mechanics and Engineering, 194(30–33):3213–3243, 2005

    work page 2005

  50. [50]

    T. Rees, H. S. Dollar, and A. J. Wathen. Optimal solvers for pde-constrained optimization.SIAM Journal on Scientific Computing, 32(1):271–298, 2010

    work page 2010

  51. [51]

    Bradbury, R

    J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang. Jax: composable transformations of python+numpy programs. 2018. Software available athttp://github.com/google/jax

    work page 2018

  52. [52]

    Griewank and A

    A. Griewank and A. Walther.Evaluating Derivatives: Principles and Techniques of Algorithmic Differen- tiation. SIAM, 2008

    work page 2008

  53. [53]

    J. Bleyer. Learning elastoplasticity with implicit layers.SSRN preprint, 2025. 39

    work page 2025

  54. [54]

    Springer Berlin Heidelberg, Berlin, Heidelberg, 1984

    Clifford Truesdell.The Elements of Continuum Mechanics. Springer Berlin Heidelberg, Berlin, Heidelberg, 1984

    work page 1984

  55. [55]

    John M. Ball. Convexity conditions and existence theorems in nonlinear elasticity.Archive for Rational Mechanics and Analysis, 63(4):337–403, 1976

    work page 1976

  56. [56]

    A new family of Constitutive Artificial Neural Networks towards automated model discovery.Computer Methods in Applied Mechanics and Engineering, 403:115731, jan 2023

    Kevin Linka and Ellen Kuhl. A new family of Constitutive Artificial Neural Networks towards automated model discovery.Computer Methods in Applied Mechanics and Engineering, 403:115731, jan 2023

    work page 2023

  57. [57]

    V. Tac, A. Amiri-Hezaveh, G.N. Bechtel, Rausch M.K., F.S. Costabal, and A.B. Tepole. Benchmarking physics-informed frameworks for data-driven hyperelasticity.Computational Mechanics, 73:49–65, 2024

    work page 2024

  58. [58]

    Abdolazizi, Kevin Linka, and Christian J

    Kian P. Abdolazizi, Kevin Linka, and Christian J. Cyron. Viscoelastic constitutive artificial neural net- works (vCANNs) – A framework for data-driven anisotropic nonlinear finite viscoelasticity.Journal of Computational Physics, 499:112704, 2023

    work page 2023

  59. [59]

    Boyd and Lieven Vandenberghe.Convex optimization

    Stephen P. Boyd and Lieven Vandenberghe.Convex optimization. Cambridge University Press, 2004

    work page 2004

  60. [60]

    Patrick Wilber and Jay R

    J. Patrick Wilber and Jay R. Walton. The Convexity Properties of a Class of Constitutive Models for Biological Soft Issues.Mathematics and Mechanics of Solids, 7(3):217–235, 2002

    work page 2002

  61. [61]

    Zico Kolter

    Brandon Amos, Lei Xu, and J. Zico Kolter. Input Convex Neural Networks. InProceedings of the 34th International Conference on Machine Learning. PMLR, 2017

    work page 2017

  62. [62]

    Kevin Linka, Sarah R. St. Pierre, and Ellen Kuhl. Automated model discovery for human brain using Constitutive Artificial Neural Networks.Acta Biomaterialia, 160:134–151, apr 2023

    work page 2023

  63. [63]

    L. R. Herrmann. Elasticity equations for incompressible and nearly-incompressible materials by a varia- tional theorem.AIAA Journal, 3(10):1896–1900, 1965

    work page 1900

  64. [64]

    J. C. Simo and R. L. Taylor. Quasi-incompressible finite elasticity in principal stretches. continuum basis and numerical algorithms.Computer Methods in Applied Mechanics and Engineering, 85(3):273–310, 1991

    work page 1991

  65. [65]

    Dassault Systèmes Simulia Corp., Providence, RI, USA, 2024

    Dassault Systèmes Simulia Corp.Abaqus Theory Guide. Dassault Systèmes Simulia Corp., Providence, RI, USA, 2024. Section on hybrid incompressible solid elements

    work page 2024

  66. [66]

    Automated derivation of the adjoint of high-level transient finite element programs.SIAM Journal on Scientific Computing, 35(4):C369–C393, 2013

    Patrick E Farrell, David A Ham, Simon W Funke, and Marie E Rognes. Automated derivation of the adjoint of high-level transient finite element programs.SIAM Journal on Scientific Computing, 35(4):C369–C393, 2013

    work page 2013

  67. [67]

    A finite element formulation for nonlinear incompressible elastic and inelastic analysis.Computers & Structures, 26(1-2):357–409, 1987

    Theodore Sussman and Klaus-Jürgen Bathe. A finite element formulation for nonlinear incompressible elastic and inelastic analysis.Computers & Structures, 26(1-2):357–409, 1987

    work page 1987

  68. [68]

    Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber.Polymer testing, 27(6):765–777, 2008

    Luc Meunier, Grégory Chagnon, Denis Favier, Laurent Orgéas, and Pierre Vacher. Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber.Polymer testing, 27(6):765–777, 2008

    work page 2008

  69. [69]

    McCulloch, Skyler R

    Jeremy A. McCulloch, Skyler R. St. Pierre, Kevin Linka, and Ellen Kuhl. On sparse regression, Lp- regularization, and automated model discovery.International Journal for Numerical Methods in Engineer- ing, 125(14):e7481, jul 2024

    work page 2024

  70. [70]

    Lecun, L

    Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998

    work page 1998

  71. [71]

    Mechanical mnist: A benchmark dataset for mechanical metamodels.Extreme Mechanics Letters, 36:100659, 2020

    Emma Lejeune. Mechanical mnist: A benchmark dataset for mechanical metamodels.Extreme Mechanics Letters, 36:100659, 2020

    work page 2020

  72. [72]

    Mechanical mnist challenge dataset.https://www.manuelrausch.com/mnist-challenge, 2025

    Manuel Rausch, Adrian Buganza Tepole, Jan Fuhg, and Francisco Sahli Costabal. Mechanical mnist challenge dataset.https://www.manuelrausch.com/mnist-challenge, 2025. Accessed: 2026-05-15

    work page 2025

  73. [73]

    V. Tac, A. Amiri-Hezaveh, G.N. Bechtel, Rausch M.K., F.S. Costabal, and A.B. Tepole. Fully data-driven inverse characterization of heterogeneous materials with hyper-network neural odes.npj Computational Materials, 12(165), 2026. 40

    work page 2026

  74. [74]

    Handling vanishing gradient problem using artificial derivative

    Zheng Hu, Jiaojiao Zhang, and Yun Ge. Handling vanishing gradient problem using artificial derivative. IEEE Access, 9:22371–22377, 2021

    work page 2021

  75. [75]

    Holthusen and E

    H. Holthusen and E. Kuhl. A complement to neural networks for anisotropic inelasticity at finite strains. Computer Methods in Applied Mechanics and Engineering, 2026

    work page 2026

  76. [76]

    Holthusen, P

    H. Holthusen, P. Steinmann, and E. Kuhl. A convex route to thermomechanics: Learning internal energy and dissipation.arXiv preprint, 2026

    work page 2026

  77. [77]

    K. P. Abdolazizi, K. Linka, and C. J. Cyron. Viscoelastic constitutive artificial neural networks (vCANNs) – a framework for data-driven anisotropic nonlinear finite viscoelasticity.Journal of Computational Physics, 499:112704, 2024

    work page 2024

  78. [78]

    Kalina, Jörg Brummund, Wai Ching Sun, and Markus Kästner

    Max Rosenkranz, Karl A. Kalina, Jörg Brummund, Wai Ching Sun, and Markus Kästner. Viscoelasticty with physics-augmented neural networks: model formulation and training methods without prescribed internal variables.Computational Mechanics, 74(6):1279–1301, dec 2024

    work page 2024

  79. [79]

    Kalina, Jörg Brummund, WaiChing Sun, and Markus Kästner

    Karl A. Kalina, Jörg Brummund, WaiChing Sun, and Markus Kästner. Neural networks meet anisotropic hyperelasticity: A framework based on generalized structure tensors and isotropic tensor functions.Com- puter Methods in Applied Mechanics and Engineering, 437:117725, mar 2025

    work page 2025

  80. [80]

    Flaschel, T

    M. Flaschel, T. Hastie, and E. Kuhl. Non-smooth optimization meets automated material model discovery. arXiv preprint arXiv:2507.10196, 2025

    work page internal anchor Pith review arXiv 2025

Showing first 80 references.