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A recurrent network architecture built from two input-convex Kolmogorov–Arnold Networks converts stress–strain data from material tests into closed-form symbolic expressions for the elastic and inelastic potentials, including temperature ef

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2026-08-02 22:13 UTC pith:W6ET3K54

load-bearing objection A useful, thermodynamically sound extension of CKANs to inelasticity, but the synthetic recovery test quietly shows the 3D potentials are not identifiable from uniaxial data, so the 'discovery' claim is overstatement as written. the 3 major comments →

arxiv 2602.17750 v3 pith:W6ET3K54 submitted 2026-02-19 cond-mat.mtrl-sci cs.AIphysics.comp-ph

Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material models

Chenyi Ji , Kian P. Abdolazizi , Hagen Holthusen , Christian J. Cyron , Kevin Linka This is my paper
classification cond-mat.mtrl-sci cs.AIphysics.comp-ph MSC 74A2074B2074D1068T07
keywords Kolmogorov-Arnold Networksconstitutive modelinginelasticityviscoelasticitysymbolic regressionfinite strainsmodel discoverytemperature-dependent materials
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 introduces inelastic Constitutive Kolmogorov–Arnold Networks (iCKANs), a recurrent neural architecture for automated material-law discovery. The central claim is that time-resolved stress–strain data can be turned into closed-form symbolic expressions for the elastic free-energy potential and the inelastic dissipation potential, with thermodynamic consistency enforced by construction rather than by post-hoc correction. If this holds, material models that currently require expert derivation could instead be learned directly from experimental data, with explicit expressions for how non-mechanical factors such as temperature enter the response. The authors demonstrate the workflow on synthetic data and on experimental viscoelastic VHB 4910 and VHB 4905 polymers, producing compact formulas that reproduce loading–unloading curves at multiple strain rates.

Core claim

iCKANs express the elastic potential and the inelastic dissipation potential as partially input-convex KANs whose B-spline activations are trained through a recurrent explicit (or implicit) time-integration scheme. Convexity of the second-layer activations, together with an H-operator that shifts the output to be convex, nonnegative, and zero at zero stress, guarantees the dissipation inequality without extra loss terms. After L1 sparsification, the trained B-splines are replaced by symbolic functions, yielding closed-form potentials such as exponential combinations of modified invariants. On VHB 4910, two parallel iCKAN branches trained only at maximum stretch 3.0 reproduce experiments at s

What carries the argument

The carrying object is the partially input-convex Kolmogorov–Arnold Network: a KAN whose B-spline activation functions are constrained to be convex and non-decreasing for the mechanical invariant inputs, while remaining unconstrained for auxiliary features such as temperature. For the inelastic potential, an H-operator transforms the convex output so that it becomes convex, nonnegative, and zero-valued with zero gradient at zero stress, providing a built-in threshold region. The two potentials are coupled through the multiplicative decomposition with co-rotated intermediate configuration, and the inelastic evolution equation is integrated with an exponential map, with an optional liquid time

Load-bearing premise

The load-bearing assumption is that the true three-dimensional material response can be identified from uniaxial, one-dimensional tests; if many different three-dimensional laws produce the same uniaxial stress–strain curves, the discovered symbolic potentials are just one of many possible fits and may fail under biaxial or shear-dominated loading.

What would settle it

Train an iCKAN on the uniaxial VHB 4910 dataset as described, then test the symbolified model against an independent biaxial or pure-shear experiment on the same material at matching strain rates: a systematic deviation in predicted stress would show that the uniaxial data do not uniquely determine the inferred three-dimensional potentials.

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

If this is right

  • Closed-form, thermodynamically consistent potentials could be discovered automatically from standard uniaxial test data, removing the need to hand-guess constitutive forms for new viscoelastic materials.
  • Once symbolified, the discovered potentials are cheap closed-form expressions, so finite element deployment adds no neural-network evaluation overhead.
  • Non-mechanical features such as temperature enter as explicit functions inside the potentials, letting one model cover a range of service conditions rather than re-fitting per condition.
  • The parallel-branch construction reproduces multiple relaxation mechanisms, capturing rate-dependent viscoelastic behavior even when training data come from a single maximum stretch level.
  • Built-in convexity provides physically consistent extrapolation beyond the training deformation range, as demonstrated in the synthetic benchmark.

Where Pith is reading between the lines

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

  • Because the experimental demonstrations use only uniaxial data, the discovered three-dimensional potentials may be non-unique: many 3D potentials can agree on uniaxial stress–strain curves, so a biaxial or shear test would be the real check of whether the symbolified law is the material law rather than one representative of an equivalence class.
  • The temperature feature is currently added only to the elastic potential; the same machinery could be applied to the inelastic potential, which would let the model learn temperature-dependent relaxation timescales that the present formulation leaves implicit.
  • The second temperature activation required a manually chosen piecewise quadratic fit with a breakpoint at 40 °C, suggesting that automated piecewise symbolification may be needed when features have non-smooth or threshold-like effects on material response.
  • The H-operator's zero-stress interval acts like a yield threshold, so the same architecture could plausibly identify plasticity-style models with an elastic domain directly from mechanical data, though the paper only demonstrates viscoelasticity.

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

3 major / 3 minor

Summary. The manuscript introduces inelastic Constitutive Kolmogorov-Arnold Networks (iCKANs), a recurrent neural-network architecture that combines the generalized standard material framework at finite strains with partially input-convex KANs for the elastic and inelastic potentials. The authors enforce thermodynamic consistency architecturally, integrate the inelastic evolution equation explicitly or implicitly, and then convert the trained B-spline activations into closed-form symbolic expressions. The method is demonstrated on a synthetic viscoelastic dataset and on experimental VHB 4910 and VHB 4905 data, the latter including temperature as an additional non-mechanical feature. The central claim is that iCKANs can automatically discover interpretable, closed-form elastic and inelastic potentials directly from material-testing data.

Significance. If substantiated, the paper would constitute a meaningful step toward automated, interpretable constitutive model discovery for inelastic solids: the thermodynamic scaffolding is standard and physically sound, the partially input-convex KAN construction with the H-operator postprocessing is a plausible way to enforce convexity, the explicit/implicit time-integration variants are carefully described, and the use of external experimental data for VHB polymers is a strength. The paper also shows predictive accuracy on the uniaxial training and validation sets. However, the central 'discovery' claim is currently not supported by the evidence: the synthetic recovery test in §5.1 produces symbolic potentials that differ from the known ground truth by including spurious invariant terms, and the experimental demonstrations are limited to uniaxial loading without multiaxial or uniqueness validation. The significance is therefore conditional on either establishing identifiability of the 3D potentials from the available data or substantially reframing the claims as model identification of one admissible potential.

major comments (3)
  1. [§5.1, Fig. 7c] The synthetic experiment is presented as verification but does not recover the generating potentials. The ground truth is ψ=0.5·Î1+0.5·Î3 and ω=0.1·(ÎΣ1)^2+(ĴΣ2)^2. The symbolified explicit results are ψ=a·Î1+b·Î2+c·Î3 with (a,b,c)=(0.0,0.2194,0.5027) and ω=a·(ÎΣ1)^2+b·(ĴΣ2)^2+c·(ĴΣ3)^2 with (0.109,0.9394,0.4286); the implicit results are similar. The recovered expressions contain spurious Î2 and Ĵ3 terms, and the coefficients of the correct terms deviate substantially. Because the training data are uniaxial stress histories, many different 3D invariant potentials can produce the same uniaxial response; this experiment therefore demonstrates non-identifiability rather than successful discovery. The abstract's claim that the method translates data into the 'corresponding' potential functions is not supported. A formal identifiability analysis or multiaxial validation, or a careful restate
  2. [§5.2.1] The VHB 4910 experimental model is trained only on uniaxial data at F11,max=3.0 and validated only on other uniaxial stretch levels. No multiaxial validation is performed. The same identifiability issue as in §5.1 therefore applies: the symbolified elastic and inelastic potentials are one of many possible 3D potentials that fit the uniaxial training data, and their extension to multiaxial loading is unverified. Since the paper's central contribution is the symbolic material law rather than merely a uniaxial curve fit, this missing validation is load-bearing.
  3. [§5.2.2] The temperature-dependent VHB 4905 example is not fully automated as claimed. The activation 2g(θ) is 'manually approximated' using two piecewise quadratic functions with C1 continuity enforced by hand, and the temperature feature is included only in the elastic potentials and deliberately omitted from the inelastic potentials 'due to network robustness'. The manual piecewise symbolification contradicts the automated-discovery claim, and omission of temperature from the inelastic branch means the model cannot capture temperature-dependent inelastic evolution. At minimum, the manual step should be stated as a limitation and justified, or an automatic piecewise symbolification procedure should be provided.
minor comments (3)
  1. [Figure 9b] The table for the VHB 4910 model lists 1ψ, 1ω, and 2ψ but does not list the second inelastic potential 2ω. Since the model consists of two parallel iCKAN branches, the symbolic form of 2ω is part of the discovered constitutive law and should be reported for reproducibility.
  2. [B.3] Code and data availability is deferred to acceptance. Given that the central claims involve symbolic extraction and reproducibility of the fitted potentials, releasing the implementation and the processed datasets would strengthen the paper.
  3. [Throughout] There are numerous typographical errors, e.g., 'listes' in Fig. 9, 'dotts' in Fig. 11, 'sastisfactory', 'Therfore', 'proove', 'funciton' in Appendix A.1, and inconsistent hyphenation. A careful language edit is needed.

Circularity Check

0 steps flagged

No significant circularity: external experimental data and held-out validation carry the central claim; self-cited framework is parameter-free prior work, and the synthetic non-uniqueness is an identifiability limitation, not a circular reduction.

full rationale

The derivation chain is self-contained rather than circular. The central experimental validation in Section 5.2 uses independent literature data (Hossain et al. 2012 for VHB 4910; Liao et al. 2020 for VHB 4905), trains on one loading level or temperature subset, and tests on held-out stretch levels and temperatures (e.g., VHB 4910 is trained only at F11,max=3.0 and validated at 1.5, 2.0, 2.5; VHB 4905 has separate training/testing panels in Figure 11). Nothing in the stress loss (Eq. 31) or the explicit update (Eq. 29) is a fitted constant renamed as a prediction. The synthetic experiment in Section 5.1 generates data from known closed-form potentials, trains on a subset (tensile loading and relaxation, t_load={0.2,0.5}s), and then evaluates the symbolified model on the full dataset including compression and t_load=1.0s; this is a recovery and extrapolation check, not a hidden re-use of the targets. The fact that the recovered expressions in Figure 7c contain spurious invariant terms and do not match the generating formulas exactly indicates under-identification of 3D potentials from uniaxial stress paths, which is a correctness or identifiability limitation, not equivalence by construction. The thermodynamic framework (co-rotated intermediate configuration, dual inelastic potential) is taken from Holthusen et al. prior publications, but those are parameter-free published assumptions with stated proofs, not a uniqueness theorem invoked to forbid alternatives. Self-citation appears but is not load-bearing in a circular way. The score is therefore 0.

Axiom & Free-Parameter Ledger

9 free parameters · 9 axioms · 0 invented entities

The framework introduces no new physical entities such as particles, forces, or dimensions. The H-operator and partially input-convex KAN are mathematical/architectural constructions, not invented physical objects. The free parameters are the fitted coefficients of the discovered symbolic potentials; the axioms are the constitutive-modeling assumptions inherited from prior co-authored work and the paper's data-identifiability premise.

free parameters (9)
  • Synthetic ψ coefficients a,b,c (explicit/implicit) = 0, 0.2194, 0.5027 / 0.2722, 0.1093, 0.5255
    Fitted by iCKAN training; differ from the true Neo-Hookean coefficients 0.5,0.5, showing non-uniqueness.
  • Synthetic ω coefficients a,b,c (explicit) = 0.109, 0.9394, 0.4286
    Fitted by training; includes an extra (J3)^2 term not present in the generating formula.
  • VHB4910 1ψ coefficients a,b,c = 34.654, 0.1204, 3.7902
    Fitted material parameters in the discovered elastic potential for branch 1.
  • VHB4910 1ω coefficients a,b,c = 0.0195, 0.0002, 0.036
    Fitted material parameters in the discovered inelastic potential for branch 1.
  • VHB4910 2ψ coefficients a,b,c,d = 2.5911, 1.8997, 68.73, 0.0299
    Fitted material parameters in the discovered elastic potential for branch 2.
  • VHB4905 1ψ coefficients a,b,c and temperature g(θ) = 0.0349, 0.0855, 0.0864; 10.0746, 0.5239, 0.0111
    Fitted by training; temperature feature enters only through g(θ).
  • VHB4905 1ω coefficients a,b,c,d,e = 31.918, 0.1077, 4.7, 0.0382, 10.0
    Fitted material parameters in the discovered inelastic potential for branch 1.
  • VHB4905 2ψ coefficients a,b,c,d and piecewise 2g(θ) = 0.319, 0.0261, 0.0921, 10.0; two quadratic polynomials
    The temperature function 2g(θ) was manually approximated by two piecewise quadratics with C1 continuity, a post-hoc fit rather than an automated output.
  • KAN grid-range extension factor = 20%
    Chosen by hand for all experiments; the effective inputs are generated during the recurrent forward pass and their range is not known a priori.
axioms (9)
  • domain assumption Multiplicative decomposition F = Fe Fi with a co-rotated intermediate configuration
    Section 2; taken from Holthusen et al. 2023/2024b. If this decomposition does not hold for the material, the whole elastic/inelastic split is invalid.
  • domain assumption Generalized Standard Materials framework: elastic potential ψ and dual inelastic potential ω with evolution D_i = ∂ω/∂Σ
    Section 2; cited to Halphen and Nguyen 1975 and Holthusen et al. 2026. This is the model class the network is forced into.
  • standard math Convex, zero-at-origin, nonnegative ω is sufficient for non-negative dissipation
    Section 2; standard thermodynamic result cited to Germain et al. 1983.
  • domain assumption The modified stress invariants (I1, sqrt(J2), cuberoot(J3)) preserve convexity and guarantee non-negative dissipation
    Equation (5) and footnote; the convexity of the third modified invariant is asserted via Holthusen et al. 2026, not proved in this paper.
  • standard math Polyconvexity of the modified elastic invariants Î1, Î2, Î3
    Equation (22); cited to Hartmann and Neff 2003. Needed so the input-convex elastic KAN yields polyconvex ψ(F).
  • standard math Kolmogorov-Arnold representation with B-spline activations can represent the relevant potentials
    Section 3.1; the KAT is a standard existence theorem, and the approximation capacity of B-spline KANs is assumed.
  • ad hoc to paper Incompressibility of the VHB experimental data
    Section 5.2: the experimental data are one-dimensional, so volumetric modes are assumed absent and the materials are treated as incompressible.
  • ad hoc to paper Identifiability of 3D potentials from uniaxial experimental data
    Section 5.2: training on uniaxial stress-strain curves is assumed sufficient to determine a valid 3D inelastic potential; this is the weakest load-bearing premise.
  • domain assumption Explicit exponential integrator for the evolution equation is stable and accurate for the tested ranges
    Section 3.2.4, Equation (29): stability is controlled by time step selection, and the explicit scheme is used for inference and extrapolation.

pith-pipeline@v1.3.0-alltime-deepseek · 26887 in / 21210 out tokens · 207277 ms · 2026-08-02T22:13:18.814026+00:00 · methodology

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read the original abstract

A key problem of solid mechanics is the identification of the constitutive law of a material, that is, the relation between strain history and stress. Machine learning has lead to considerable advances in this field lately. Here we introduce inelastic Constitutive Kolmogorov-Arnold Networks (iCKANs). This novel artificial neural network architecture can discover in an automated manner symbolic constitutive laws describing both the elastic and inelastic behavior of materials. That is, it can translate data from material testing into corresponding elastic and inelastic potential functions in closed mathematical form. We demonstrate the advantages of iCKANs using both synthetic data and experimental data of the viscoelastic polymer materials VHB 4910 and VHB 4905. The results demonstrate that iCKANs accurately capture complex viscoelastic behavior while preserving physical interpretability. It is a particular strength of iCKANs that they can process not only mechanical data but also arbitrary additional information available about a material (e.g., about temperature-dependent behavior). This makes iCKANs a powerful tool to discover in the future also how specific processing or service conditions affect the properties of materials.

Figures

Figures reproduced from arXiv: 2602.17750 by Chenyi Ji, Christian J. Cyron, Hagen Holthusen, Kevin Linka, Kian P. Abdolazizi.

Figure 1
Figure 1. Figure 1: Inelastic Constitutive Kolmogorov-Arnold Network (iCKAN) pipeline for automated interpretable model [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Multiplicative split of the deformation gradient into elastic and inelastic part, including the non [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Demonstration of the postprocessing of a convex and non-decreasing function [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representation of the iCKAN formulation in the form of a Maxwell model. The elastic potential [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Explicit iCKAN architecture at timestep t. The inputs at each step are the current deformation gradient and time increment (Ft, ∆t) together with the state variables (Ct−1, Ui,t−1) from the previous step. The KAN models for the elastic potential ψ and the inelastic potential ω are evaluated to update the state variables according to Equation 29. The updated state is then propagated to the next time step. I… view at source ↗
Figure 6
Figure 6. Figure 6: Symbolification of the activation functions of the trained iCKAN on synthetic data. Black curves de [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Results of the symbolified iCKAN on synthetic compressible material data using (a) the explicit time [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: KAN architectures of discovered model for VHB4910 polymer. (b) Rheological model consisting two [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Discovered model for VHB4910 polymer. (a) Predictions by the symbolified trained iCKAN under [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Discovered model for VHB4905 thermo polymer [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (a) Training set and (b) validation set of discovered model for the experimental data of VHB 4905 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Discovered elastic and inelastic potential functions for VHB 4905 thermo polymer. [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Discovering Thermodynamically Admissible Dissipation Potentials via Grammar-Based Symbolic Regression

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    A convexity-preserving grammar enables symbolic regression to discover thermodynamically admissible dissipation potentials for generalized standard materials from noisy data.

  2. LLM-driven design of physics-constrained constitutive models: two agents are better than one

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    A Creator-Inspector multi-agent LLM pipeline for constitutive artificial neural networks increases the rate of models satisfying all nine physical constraints to 100% or 56% depending on the LLM backbone.

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