Pith. sign in

REVIEW 2 major objections 5 minor 59 references

Learning Memory and Material Dependent Constitutive Laws

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read A recurrent neural operator can approximate the exact homogenized viscoelastic constitutive law to any prescribed accuracy.

desk verdict The FNM-RNO architecture is a useful contribution and the numerics are convincing, but the main universal approximation theorem has a genuine proof gap that needs fixing before the theory can be cited. read the letter →

arxiv 2502.05463 v2 pith:MMYRWZZS submitted 2025-02-08 math.NA cs.LGcs.NA

Kaushik Bhattacharya , Lianghao Cao , George Stepaniants , Andrew Stuart , Margaret Trautner This is my paper · ORCID
classification math.NAcs.LGcs.NA MSC 35B2765M6068T0774D0574D1074Q1074Q15
keywords constitutivemodelinghomogenizationmemorymicrostructureKelvin-Voigtviscoelasticityneuraloperatoruniversalapproximation
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

Heterogeneous materials are usually modeled by homogenization, which replaces fine microstructure with an effective macroscale law, but the cell problems that define that law are expensive and must be redone for every new material. This paper constructs a recurrent neural operator, FNM-RNO, that takes both the strain history and the microstructure function as inputs and outputs the averaged stress. The central theoretical result is a universal approximation theorem: in one-dimensional Kelvin-Voigt viscoelasticity, such a network can approximate the exact homogenized constitutive law uniformly to any prescribed error over a large class of microstructures and strain inputs. If correct, the learned law can be deployed in macroscale simulations and applied to new microstructures without retraining, and the numerical experiments show it also works for elasto-viscoplastic materials beyond the theory.

What carries the argument

The central object is the FNM-RNO, a neural differential equation with internal state vector $\xi(t)$, stress output $\sigma(t)=F_{\mathrm{FNM}}(\epsilon(t),\dot{\epsilon}(t),\xi(t);M)$, and dynamics $\dot{\xi}(t)=G_{\mathrm{FNM}}(\epsilon(t),\xi(t);M)$, where $M$ is the microstructure function and $F_{\mathrm{FNM}},G_{\mathrm{FNM}}$ are Fourier Neural Mappings. The proof machinery has three parts: the cell problem is Lipschitz in the microstructure, so small changes in the material fields produce controlled changes in stress; any bounded-variation microstructure can be replaced by a piecewise-constant one whose homogenized memory kernel is a finite sum of exponentials, reducing the constitutive law to a linear ODE with continuous coefficients; and Fourier Neural Mappings universally approximate the ODE right-hand sides on compact sets, which, combined with Gronwall's inequality, yields the theorem.

What would settle it

Train the FNM-RNO on the piecewise-constant Kelvin-Voigt cell problem at decreasing error tolerances and measure the Lipschitz constant of the learned right-hand side with respect to the internal state; if that constant grows faster than the reciprocal of the training error, the Gronwall factor in the proof cannot be driven to zero and the uniform approximation bound of Theorem 4.7 would fail for long time horizons.

Watch

Extended reading notes

Core claim

The central claim, Theorem 4.7, is that for any tolerance $e>0$ there exist Fourier Neural Mappings $F_{\mathrm{FNM}}$ and $G_{\mathrm{FNM}}$ such that the FNM-RNO map uniformly approximates the exact homogenized Kelvin-Voigt constitutive law on the set of microstructures with bounded variation and bounded strain trajectories. The proof isolates three properties: Lipschitz continuity of the cell problem in the microstructure, the fact that piecewise-constant microstructures produce a memory kernel that is a finite sum of exponentials and hence a finite-dimensional linear ODE, and the universal approximation property for Fourier Neural Mappings. The theorem is stated in one-dimensional linear viscoelasticity; the numerical section extends the architecture to continuous high-memory microstructures and to nonlinear elasto-viscoplasticity.

Load-bearing premise

The proof assumes that making the neural-network approximation of the cell-problem dynamics arbitrarily accurate does not make it so sensitive to the internal state that small errors are amplified beyond control over the time interval; the paper does not establish this joint scaling.

Editorial extensions

If this is right

  • A single trained FNM-RNO can predict stress for microstructures outside its training set, as long as they satisfy the bounded-variation assumptions, with error uniformly controlled by the theorem.
  • The model can be embedded in macroscale simulations, replacing repeated cell-problem solves; in the paper's experiments its displacement error is comparable to a multiscale simulation with 20 cells per wavelength.
  • The architecture is discretization agnostic: predictions remain accurate when evaluated at spatial and temporal resolutions different from the training data.
  • The same architecture learns elasto-viscoplastic constitutive laws with mean relative errors around 1-3% on stress and plastic strain, even though the theoretical guarantee is for linear viscoelasticity.
  • Including at least a few internal variables substantially reduces error compared to a memoryless stress response, confirming that the recurrent structure captures the homogenized memory kernel.

Reading between the lines

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

  • If the Gronwall factor is the bottleneck, the uniform guarantee degrades with time horizon, so long-time deployment may need a stability analysis of the recurrent operator itself.
  • The same internal-variable construction could extend to other history-dependent homogenized laws, such as thermo-viscoelastic or poroelastic composites, wherever a Markovian internal-state ODE representation exists.
  • A direct empirical check of the theorem's assumption would be to train FNM-RNOs at decreasing error thresholds and measure the sensitivity of the learned right-hand side to the internal state; if that sensitivity grows too fast, the uniform approximation claim would need a modified architecture or a different error decomposition.
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

2 major / 5 minor

Summary. The paper proposes a recurrent neural operator architecture, FNM-RNO, for learning homogenized constitutive laws that depend both on strain history and on the material microstructure, and it analyzes this architecture in the setting of one-dimensional Kelvin-Voigt viscoelasticity. The theoretical part establishes Lipschitz continuity of the cell-problem solution map with respect to material coefficients, proves that continuous bounded-variation microstructures can be approximated by piecewise-constant ones, and states a universal approximation theorem (Theorem 4.7) claiming that FNM-RNO can uniformly approximate the exact homogenized stress-strain map. The numerical part trains the model on viscoelastic and elasto-viscoplastic cell-problem data and demonstrates accurate stress prediction, generalization from piecewise-constant to continuous microstructures, robustness to spatial and temporal resolution changes, and deployment in macroscale simulations.

Significance. If Theorem 4.7 is established, the paper would provide the first universal approximation guarantee for a microstructure-dependent, memory-dependent learned constitutive law, and it would justify the proposed architecture beyond heuristics. The supporting analytical results are valuable in themselves: the Lipschitz regularity of the cell-problem map is carefully proved, the piecewise-constant approximation result is clean, and the explicit closed-form expression for the Prony coefficients in Lemma 3.4 is a useful contribution. The numerical study is extensive, including 2,500-sample test sets, two material families for viscoelasticity, an elasto-viscoplastic extension, and macroscale deployment. The main caveat is that the proof of the central approximation theorem contains a load-bearing gap concerning the Lipschitz constant of the learned dynamics; this needs to be repaired before the universal approximation claim can be accepted.

major comments (2)
  1. [§4.3, Eqs. (4.32)–(4.34)] The proof of Theorem 4.7 assumes that the term (e_G/L_G) exp(L_G T) can be made arbitrarily small by choosing e_G sufficiently small. Proposition C.3 guarantees only the existence of an FNM GFNM with uniform approximation error below e_G; it provides no control on L_G, the Lipschitz constant of GFNM in the hidden-state variable ξ. Universal-approximation constructions can have Lipschitz constants that grow as the approximation error shrinks, in which case the Gronwall factor in (4.32) may diverge for every positive T. The same issue affects the a priori bound on ξ_RNO in (4.33), which is used to define the compact set on which FFNM is approximated. As written, the claimed bound (4.34) does not follow from the stated results; a repair would require an explicit bounded-Lipschitz FNM construction for the affine map G_pc, or a different argument that controls the error-versus-Lipschitz trade-off.
  2. [§4.3, Theorem 4.7] The proof implicitly identifies the internal-state dimension L of the FNM-RNO with the dimension n(e/2) of the hidden state ξ_pc in the piecewise-constant model Ψ_pc. This identification is not stated, and the bounds (4.30)–(4.34) use √L for both ξ_pc and the linear term ⟨1, ξ_RNO − ξ_pc⟩. If L and n(e/2) differ, those expressions compare vectors of different dimensions. Setting L = n(e/2) explicitly in the theorem statement would fix this, but it is a necessary clarification for the proof to be coherent.
minor comments (5)
  1. [Theorem 4.7] The opening sentence of Theorem 4.7 states a pointwise claim for any E, ν, and ϵ satisfying Assumptions 1, but the proof establishes a uniform statement over the bounded-variation classes described in (4.28). Please align the wording with the uniform claim.
  2. [Definition 2.4] The symbol T is used both for the time interval [0, T] and for the number of Fourier layers in the FNM. This is confusing in definitions such as the layer index t ∈ [T ]; a different symbol for the number of layers would improve readability.
  3. [Figure 3] The stress-prediction panels in Figure 3 are repeated across columns in a way that makes it difficult to see which column corresponds to which model. Please simplify the layout or use clearer separation between the true, FNM-RNO, and no-memory curves.
  4. [§5.3] The phrase 'discretization agnostic' is stronger than what the experiments show: the results in Figure 4 indicate sensitivity to temporal resolution, as the authors acknowledge. A phrase such as 'robust to changes in resolution' would be more accurate.
  5. [Appendix D] Appendix D is an honest and useful discussion, but its main message—that training without the penalty term can give smaller relative L2 error while having much larger relative L∞ error—should be summarized in the main text, since it directly informs the loss-function choice in (5.3).

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation chain found; the main issue is an unproven error-versus-Lipschitz trade-off in the proof of Theorem 4.7, which is a correctness gap rather than a circular reduction.

full rationale

The paper's architecture is deliberately patterned on the homogenized Kelvin-Voigt internal-variable ODE, but that is motivation, not circularity: the target map Ψ0 is defined independently by the cell problem (3.3), and the universal approximation theorem gives an existence statement over a class of FNM approximants. The piecewise-constant exact representation (Proposition 3.3) is quoted from prior work [6] by the same group, but it is a parameter-free mathematical theorem with stated assumptions that do not include the FNM-RNO result, so it is independent support under the review rules. Similarly, the FNO universal approximation input (Proposition C.2, adapted from [27]) is an external theorem, not fitted to the data. The proof of Theorem 4.7 does contain a genuine gap at equations (4.32)-(4.34): Proposition 4.6/C.3 provides GFNM with pointwise error eG but gives no control on the Lipschitz constant LG as eG shrinks, so the Gronwall factor (eG/LG) exp(LG T) is not shown to be small; however, this is an unproved scaling assumption, not a circular identification of a prediction with an input. No fitted parameter is renamed as a prediction, and no conclusion reduces by construction to a definition or a self-citation.

Assumptions & free parameters 5 free parameters · 7 assumptions · 1 invented entities

The framework rests on the standard homogenization representation of Kelvin-Voigt materials, the exact finite-exponential memory kernel for piecewise-constant microstructures from the authors' prior work, and the universal approximation property of Fourier neural mappings. The main unproved background assumptions are the compactness of the admissible material class and, critically, the boundedness of the FNM Lipschitz constant in the approximation used for Theorem 4.7.

free parameters (5)
  • Internal state dimension L = 5 for viscoelastic experiments; scalar for elasto-viscoplastic experiments
    Chosen by hand from the set {1,3,5,10,15,20,25}. Larger L did not improve accuracy and made training harder; the theory requires at least L-1 internal variables for piecewise-constant materials with L pieces.
  • Number of Fourier modes per convolution layer = 4 (viscoelastic), 2 (elasto-viscoplastic)
    Architecture hyperparameter; no selection rule given.
  • Fourier layer count and hidden width = 3 Fourier layers, 32 channels
    Fixed architecture choices in Section 5.2.
  • Loss penalty coefficient for GFNM(0,0)=0 = 1 (term added without scaling)
    Ad hoc constraint introduced because without it models violate the equilibrium GFNM(0,0)=0 (Appendix D).
  • Training schedule = learning rate 1e-3, batch size 32, 500 epochs, cosine annealing to 1e-5
    Standard training hyperparameters; not tuned theoretically.
assumptions (7)
  • domain assumption Admissible microstructures E, nu lie in MB_{Emin,Emax} and MB_{nu_min,nu_max}; strain epsilon lies in C_{epsilon_max,epsilondot_max}.
    Assumptions 1 (Section 3.2) and used throughout; needed for cell problem well-posedness, Lipschitz bounds, and compactness.
  • standard math The homogenized Kelvin-Voigt law has the convolution form (3.12) with memory kernel K(t).
    Francfort-Suquet [19], background homogenization theorem.
  • standard math For piecewise-constant microstructures the exact law equals the linear ODE (3.20) with positive diagonal A and nonnegative b.
    Proposition 3.3 (Theorem 3.6 of [6]) and Lemma 3.4; the starting point for the piecewise-constant approximation.
  • standard math Fourier neural mappings are universal approximators for continuous maps on compact subsets of vector and function spaces.
    Proposition C.2, adapting [27], [31]; used for Propositions 4.6 and Theorem 4.7.
  • standard math The set of admissible material functions is compact in L2.
    BV(T) intersected with L-infinity(T) embeds compactly into L2(T); [5, Lemma C.1].
  • domain assumption The FNM layers satisfy the boundedness and activation-Lipschitz conditions of Assumptions 2.
    Lemma C.4 computes the FNM Lipschitz constant under these assumptions; training does not enforce them.
  • standard math The cell problem has unique solutions with the bounds in Corollary B.2.
    Proved in Appendix B using energy estimates and Gronwall's inequality.
invented entities (1)
  • Internal state vector xi(t) in FNM-RNO independent evidence
    purpose: Markovian encoding of strain history so the constitutive law is causal and discretization-agnostic.
    For piecewise-constant microstructures the exact law has a finite-dimensional internal state (Prop 3.3); in elasto-viscoplasticity the learned xi tracks averaged plastic strain up to a constant (Fig 8); the penalty term enforces xi(0)=0 equilibrium.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Learning Memory and Material Dependent Constitutive Laws." pith.science (2026). https://pith.science/paper/MMYRWZZS

@misc{pith2026250205463,
  author       = {Pith},
  title        = {Pith review of: Learning Memory and Material Dependent Constitutive Laws},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MMYRWZZS}},
  note         = {Machine review of arXiv:2502.05463}
}
read the original abstract

We propose and study a neural operator framework for learning memory- and material microstructure-dependent constitutive laws for heterogeneous materials. We work in the two-scale setting where homogenization theory provides a systematic approach to deriving macroscale constitutive laws, obviating the need to resolve complex microstructure repeatedly. However, the unit cell problems defining these constitutive models are typically not amenable to explicit evaluation. It is therefore of interest to learn constitutive models from data generated by the unit cell problem. Our proposed framework models homogenized constitutive laws with both memory- and microstructure-dependence through the use of Markovian recurrent and Fourier neural operators. The homogenization problem for Kelvin-Voigt viscoelastic materials is studied to provide firm theoretical foundations for our model. The theoretical properties of the cell problem in this Kelvin-Voigt setting motivate the proposed learning framework; and are also used to prove a universal approximation theorem for the learned macroscale constitutive model. Numerical experiments show that the proposed learning framework accurately learns memory- and microstructure-dependent viscoelastic and elasto-viscoplastic constitutive models, beyond the setting of the theory. Furthermore, we show that the learned constitutive models can be successfully deployed in macroscale simulation of material deformation for different microstructures without retraining.

Figures

Figures reproduced from arXiv: 2502.05463 by the authors.

Figure 1
Figure 1. Visualization of samples from the two datasets: piecewise-constant material [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. The distributions of the relative L 2 error on 2,500 testing samples from the PC dataset (left) and the HMC dataset (right). We visualize the errors in FNM–RNOs predictions where the trained FNM–RNOs have a varying number of internal variables. We also visualize the distribution of error given by the linear stress response without memory effects, where the response function is obtained using (3.12) with K ≡ 0. Discr… view at source ↗
Figure 4
Figure 4. The mean relative L 2 error on the PC (left) and HMC (right) testing samples at different testing resolutions for the trained FNM–RNO with 5 internal variables [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: The distributions of the relative [PITH_FULL_IMAGE:figures/full_fig_p030_5.png]
Figure 6
Figure 6. Figure 6: Visualization of macroscale solutions using different constitutive models for [PITH_FULL_IMAGE:figures/full_fig_p031_6.png]
Figure 7
Figure 7. Figure 7: The distributions of the relative L 2 error on 2,500 testing samples from the PC-EVP dataset (left) and the C-EVP dataset (right) for elasto-viscoplasticity. We consider the error in the the trained FNM–RNO predictions of (i) the averaged stress σ and (ii) the averaged…
Figure 8
Figure 8. Figure 8: Visualization of testing samples and FNM–RNO predictions (averaged stress [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]
Figure 9
Figure 9. Figure 9: The distributions of the relative L∞ error on 2,500 testing samples from the PC dataset (top) and the HMC dataset (bottom). We visualize the errors in the FNM– RNO predictions where FNM–RNOs (i) are trained with or without the penalty term in (5.3), and (ii) have a var…
Figure 10
Figure 10. Figure 10: Visualization of FNM–RNO predictions at the sample from the PC testing [PITH_FULL_IMAGE:figures/full_fig_p045_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 56 canonical work pages

  1. [1]

    Abdulle, E

    A. Abdulle, E. Weinan, B. Engquist, and E. V anden-Eijnden , The heterogeneous multi- scale method, Acta Numerica, 21 (2012), pp. 1–87

    work page 2012

  2. [2]

    Learning constitutive relations from experiments: 1. PDE constrained optimization

    A. Akerson, A. Rajan, and K. Bhattacharya , Learning constitutive relations from experi- ments: 1. pde constrained optimization , arXiv preprint arXiv:2412.02864, (2024)

    work page Pith review arXiv 2024

  3. [3]

    Allaire , Homogenization and two-scale convergence , SIAM Journal on Mathematical Analysis, 23 (1992), pp

    G. Allaire , Homogenization and two-scale convergence , SIAM Journal on Mathematical Analysis, 23 (1992), pp. 1482–1518

    1992

  4. [4]

    Bensoussan, J.-L

    A. Bensoussan, J.-L. Lions, and G. Papanicolaou , Asymptotic analysis for periodic struc- tures, vol. 374, American Mathematical Soc., 2011

    2011

  5. [5]

    Bhattacharya, N

    K. Bhattacharya, N. B. Kovachki, A. Rajan, A. M. Stuart, and M. Trautner, Learning homogenization for elliptic operators , SIAM Journal on Numerical Analysis, 62 (2024), pp. 1844–1873

    work page 2024

  6. [6]

    Bhattacharya, B

    K. Bhattacharya, B. Liu, A. Stuart, and M. Trautner, Learning markovian homogenized models in viscoelasticity, Multiscale Modeling & Simulation, 21 (2023), pp. 641–679

    work page 2023

  7. [7]

    Billington, A

    E. Billington, A. Tate, and W. Williams , The physics of deformation and flow , 1982

    work page 1982

  8. [8]

    Bishara, Y

    D. Bishara, Y. Xie, W. K. Liu, and S. Li , A state-of-the-art review on machine learning- based multiscale modeling, simulation, homogenization and design of materials , Archives of computational methods in engineering, 30 (2023), pp. 191–222

    work page 2023

Show all 59 references
  1. [9]

    Blanc and C

    X. Blanc and C. Le Bris, Homogenization Theory for Multiscale Problems: An Introduction , vol. 21, Springer Nature, 2023

    2023

  2. [10]

    Brenner and P

    R. Brenner and P. Suquet , Overall response of viscoelastic composites and polycrystals: exact asymptotic relations and approximate estimates , International Journal of Solids and Structures, 50 (2013), pp. 1824–1838

    2013

  3. [11]

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

    2018

  4. [12]

    Cioranescu and P

    D. Cioranescu and P. Donato , An introduction to homogenization , Oxford university press, 1999

    1999

  5. [13]

    B. D. Coleman and W. Noll , Foundations of linear viscoelasticity , Reviews of modern physics, 33 (1961), p. 239

    1961

  6. [14]

    B. D. Coleman and W. Noll , Recent results in the continuum theory of viscoelastic fluids , Annals of the New York Academy of Sciences, 89 (1961), pp. 672–714

    1961

  7. [15]

    Darrow and G

    D. Darrow and G. Stepaniants, A spectral theory of scalar volterra equations, arXiv preprint arXiv:2503.06957, (2025)

    2025

  8. [16]

    Dupont, A

    E. Dupont, A. Doucet, and Y. W. Teh , Augmented neural odes, Advances in neural infor- mation processing systems, 32 (2019)

    2019

  9. [17]

    Eggersmann, T

    R. Eggersmann, T. Kirchdoerfer, S. Reese, L. Stainier, and M. Ortiz , Model-free data- driven inelasticity, Computer Methods in Applied Mechanics and Engineering, 350 (2019), pp. 81–99

    2019

  10. [18]

    Ferry, Viscoelastic Properties of Polymers , vol

    J. Ferry, Viscoelastic Properties of Polymers , vol. 264, Wiley, 1980

    1980

  11. [19]

    G. A. Francfort and P. M. Suquet, Homogenization and mechanical dissipation in thermo- viscoelasticity, Archive for Rational Mechanics and Analysis, 96 (1986), pp. 265–293

    1986

  12. [20]

    J. N. Fuhg, G. Anantha Padmanabha, N. Bouklas, B. Bahmani, W. Sun, N. N. Vlassis, M. Flaschel, P. Carrara, and L. De Lorenzis , A review on data-driven constitutive laws for solids , Archives of Computational Methods in Engineering, (2024), pp. 1–43

    2024

  13. [21]

    J. N. Fuhg and 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), p. 114915

    2022

  14. [22]

    Ghavamian and A

    F. Ghavamian and A. Simone , Accelerating multiscale finite element simulations of history- dependent materials using a recurrent neural network , Computer Methods in Applied Me- chanics and Engineering, 357 (2019), p. 112594

    2019

  15. [23]

    Gross, Mathematical structure of the theories of viscoelasticity , (No Title), (1968)

    B. Gross, Mathematical structure of the theories of viscoelasticity , (No Title), (1968)

    1968

  16. [24]

    Guedes and N

    J. Guedes and N. Kikuchi , Preprocessing and postprocessing for materials based on the ho- mogenization method with adaptive finite element methods , Computer methods in applied mechanics and engineering, 83 (1990), pp. 143–198

    1990

  17. [25]

    Haghighat, S

    E. Haghighat, S. Abouali, and R. V aziri, Constitutive model characterization and discovery using physics-informed deep learning , Engineering Applications of Artificial Intelligence, 120 (2023), p. 105828

    2023

  18. [26]

    M. F. Horstemeyer and D. J. Bammann , Historical review of internal state variable theory for inelasticity , International Journal of Plasticity, 26 (2010), pp. 1310–1334

    2010

  19. [27]

    D. Z. Huang, N. H. Nelsen, and M. Trautner , An operator learning perspective on parameter-to-observable maps, Foundations of Data Science, (2024). LEARNING MEMORY & MATERIAL DEPENDENT CONSTITUTIVE LA WS 47

    2024

  20. [28]

    R. E. Jones, A. L. Frankel, and K. Johnson , A neural ordinary differential equation frame- work for modeling inelastic stress response via internal state variables , Journal of Machine Learning for Modeling and Computing, 3 (2022)

    2022

  21. [29]

    Karimi and K

    M. Karimi and K. Bhattacharya , A learning-based multiscale model for reactive flow in porous media, Water Resources Research, 60 (2024), p. e2023WR036303

    2024

  22. [30]

    J. H. Kim, D. Yang, and S. Park , Experimental validation for the interconversion between generalized kelvin–voigt and maxwell models using human skin tissues , Journal of Biome- chanics, 162 (2024), p. 111908

    2024

  23. [31]

    Kovachki, S

    N. Kovachki, S. Lanthaler, and S. Mishra , On universal approximation and error bounds for fourier neural operators , Journal of Machine Learning Research, 22 (2021), pp. 1–76

    2021

  24. [32]

    Kovachki, Z

    N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, and A. Anandkumar , Neural operator: Learning maps between function spaces with appli- cations to pdes , Journal of Machine Learning Research, 24 (2023), pp. 1–97

    2023

  25. [33]

    S. M. Kozlov, Averaging of random operators, Sbornik: Mathematics, 37 (1980), pp. 167–180

    1980

  26. [34]

    M. A. Kraus, M. Schuster, J. Kuntsche, G. Siebert, and J. Schneider, Parameter identifi- cation methods for visco-and hyperelastic material models, Glass Structures & Engineering, 2 (2017), pp. 147–167

    2017

  27. [35]

    Lahellec, R

    N. Lahellec, R. Masson, and P. Suquet , Effective thermodynamic potentials and internal variables: linear viscoelastic composites , Journal of the Mechanics and Physics of Solids, 188 (2024), p. 105649

    2024

  28. [36]

    Lahellec, R

    N. Lahellec, R. Masson, and P. Suquet , Effective thermodynamic potentials and internal variables: Particulate thermoviscoelastic composites, Journal of the Mechanics and Physics of Solids, 193 (2024), p. 105891

    2024

  29. [37]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar , Fourier neural operator for parametric partial differential equations , arXiv preprint arXiv:2010.08895, (2020)

    2020 arXiv

  30. [38]

    B. Liu, N. Kovachki, Z. Li, K. Azizzadenesheli, A. Anandkumar, A. M. Stuart, and K. Bhattacharya, A learning-based multiscale method and its application to inelastic impact problems, Journal of the Mechanics and Physics of Solids, 158 (2022), p. 104668

    2022

  31. [39]

    B. Liu, E. Ocegueda, M. Trautner, A. M. Stuart, and K. Bhattacharya, Learning macro- scopic internal variables and history dependence from microscopic models , Journal of the Mechanics and Physics of Solids, (2023), p. 105329

    2023

  32. [40]

    X. Liu, S. Tian, F. Tao, and W. Yu, A review of artificial neural networks in the constitutive modeling of composite materials , Composites Part B: Engineering, 224 (2021), p. 109152

    2021

  33. [41]

    Z. Liu, C. Wu, and M. Koishi , A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials, Computer Methods in Applied Mechanics and Engineering, 345 (2019), pp. 1138–1168

    2019

  34. [42]

    G. W. Milton, The Theory of Composites , Cambridge University Press, Cambridge, 2002

    2002

  35. [43]

    Mishra, J

    N. Mishra, J. Vondˇrejc, and J. Zeman, A comparative study on low-memory iterative solvers for fft-based homogenization of periodic media , Journal of Computational Physics, 321 (2016), pp. 151–168

    2016

  36. [44]

    Moulinec and P

    H. Moulinec and P. Suquet, A numerical method for computing the overall response of non- linear composites with complex microstructure , Computer methods in Applied Mechanics and Engineering, 157 (1998), pp. 69–94

    1998

  37. [45]

    Mozaffar, R

    M. Mozaffar, R. Bostanabad, W. Chen, K. Ehmann, J. Cao, and M. Bessa, Deep learning predicts path-dependent plasticity, Proceedings of the National Academy of Sciences, 116 (2019), pp. 26414–26420

    2019

  38. [46]

    Nikonov, A

    A. Nikonov, A. R. Davies, and I. Emri, The determination of creep and relaxation functions from a single experiment , Journal of Rheology, 49 (2005), pp. 1193–1211

    2005

  39. [47]

    Ostoja-Starzewski and J

    M. Ostoja-Starzewski and J. Zhang , Does a fractal microstructure require a fractional viscoelastic model? , Fractal and Fractional, 2 (2018), p. 12, https://doi.org/10.3390/ fractalfract2010012

    2018

  40. [48]

    Pavliotis and A

    G. Pavliotis and A. Stuart , Multiscale methods: averaging and homogenization , Springer Science & Business Media, 2008

    2008

  41. [49]

    Phillips and P

    R. Phillips and P. Rob , Crystals, defects and microstructures: modeling across scales , Cam- bridge University Press, 2001

    2001

  42. [50]

    J. R. Rice , Inelastic constitutive relations for solids: an internal-variable theory and its ap- plication to metal plasticity , Journal of the Mechanics and Physics of Solids, 19 (1971), pp. 433–455

    1971

  43. [51]

    S´anchez-Palencia, Non-homogeneous media and vibration theory, Lecture Note in Physics, Springer-Verlag, 320 (1980), pp

    E. S´anchez-Palencia, Non-homogeneous media and vibration theory, Lecture Note in Physics, Springer-Verlag, 320 (1980), pp. 57–65

    1980

  44. [52]

    Serra-Aguila, J

    A. Serra-Aguila, J. Puigoriol-Forcada, G. Reyes, and J. Menacho , Viscoelastic models 48 K. BHATTACHARYA, L. CAO, G. STEPANIANTS, A. STUART & M. TRAUTNER revisited: characteristics and interconversion formulas for generalized kelvin–voigt and maxwell models , Acta Mechanica Si...

    2019

  45. [53]

    Shanbhag , A computer program for interconversion between creep compliance and stress relaxation, Journal of Rheology, 67 (2023), pp

    S. Shanbhag , A computer program for interconversion between creep compliance and stress relaxation, Journal of Rheology, 67 (2023), pp. 965–975

    2023

  46. [54]

    Suquet , Elements of homogenization for inelastic solid mechanics , Homogenization tech- niques for composite media, 272 (1987), pp

    P. Suquet , Elements of homogenization for inelastic solid mechanics , Homogenization tech- niques for composite media, 272 (1987), pp. 193–278

    1987

  47. [55]

    L. Tartar, Memory effects and homogenization , in Mechanics and Thermodynamics of Con- tinua: A Collection of Papers Dedicated to BD Coleman on His Sixtieth Birthday, Springer, 1991, pp. 537–549

    1991

  48. [56]

    N. W. Tschoegl, The phenomenological theory of linear viscoelastic behavior: an introduction, Springer Science & Business Media, 2012

    2012

  49. [57]

    Weinan, Principles of multiscale modeling , Cambridge University Press, 2011

    E. Weinan, Principles of multiscale modeling , Cambridge University Press, 2011

    2011

  50. [58]

    Zhang and K

    Y. Zhang and K. Bhattacharya , Iterated learning and multiscale modeling of history- dependent architectured metamaterials, arXiv preprint arXiv:2402.12674, (2024)

    2024 arXiv

  51. [59]

    T. I. Zohdi and P. Wriggers , An introduction to computational micromechanics , Springer Science & Business Media, 2008

    2008

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.