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Constitutive Kolmogorov-Arnold Networks (CKANs): Combining Accuracy and Interpretability in Data-Driven Material Modeling

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Pith's one-line read Constitutive Kolmogorov–Arnold Networks turn stress-strain data into symbolic material laws without sacrificing accuracy or extrapolation.

desk verdict A genuinely useful KAN-based constitutive modeling framework with a strong same-material extrapolation demo, but the Ecoflex cross-material claim sits on a table with identical rows that needs an author explanation. read the letter →

arxiv 2502.05682 v2 pith:RVR4HJQ6 submitted 2025-02-08 physics.comp-ph cond-mat.mtrl-sci

Kian P. Abdolazizi , Roland C. Aydin , Christian J. Cyron , Kevin Linka This is my paper · ORCID
classification physics.comp-phcond-mat.mtrl-sci MSC 74B2068T07
keywords Kolmogorov-ArnoldNetworksdata-drivenmechanicsphysics-informedmachinelearningConstitutiveArtificialNeuralsoftmaterialssymbolicregressioninterpretablehyperelasticity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the long-standing trade-off between accuracy, interpretability, and extrapolation in data-driven material modeling can be removed by building Kolmogorov–Arnold Networks directly into a constitutive framework. The resulting Constitutive Kolmogorov–Arnold Networks (CKANs) learn a strain-energy function whose connection weights are univariate splines that, after sparsification, are converted one by one into compact symbolic expressions. On the classical vulcanized-rubber benchmark, a CKAN using a mixed invariant-and-stretch basis fits all three deformation modes with R² = 0.999 and, trained only on those data, predicts the independent general biaxial response of the same rubber. A separate CKAN trained on uniaxial Ecoflex data at three Shore hardness levels predicts the withheld fourth hardness level. If these results hold, hybrid constitutive modeling no longer has to choose between black-box flexibility and readable, transferable equations.

What carries the argument

The load-bearing object is the spline activation function inside a KAN layer. Instead of scalar weights, each connection is a cubic Hermite or B-spline curve with learnable control points, and partial monotonicity can be imposed per input. The CKAN arranges such activations so that their composition equals the strain energy $\Psi_{\mathrm{KAN}}$ over a chosen functional basis (principal stretches, principal invariants, modified invariants, or a mixed basis), differentiates $\Psi_{\mathrm{KAN}}$ to obtain stresses, and regularizes with L1 and entropy terms to force most activations to zero. The surviving activations are replaced by symbolic functions from a library, with affine parameters fit by constrained optimization, yielding a final closed-form strain energy. The separable Valanis–Landel ansatz with shared parameters is what keeps the principal-stretch version isotropic and tractable.

What would settle it

Compute stress-strain curves from a deliberately non-separable hyperelastic energy, such as $\Psi = \sum_\alpha f(\lambda_\alpha) + c(\lambda_1-1)(\lambda_2-1)$ with $c \neq 0$, train a principal-stretch CKAN on dense noiseless biaxial data, and check whether the loss can reach machine precision. If it cannot, the Valanis–Landel ansatz—not the data—is what limits the stretch-based CKAN.

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Extended reading notes

Core claim

The central claim is that a strain-energy function can be learned as a Kolmogorov–Arnold Network—a network in which each connection is a trainable one-dimensional spline rather than a scalar weight—and then converted into a symbolic formula without losing accuracy. Objectivity, material symmetry, incompressibility, and a stress-free reference state are built into the architecture; principal-stretch variants enforce the Valanis–Landel additive separability $\Psi = \sum_\alpha \omega_1(\lambda_\alpha) + \sum_\alpha \omega_{-1}(\lambda_\alpha^{-1})$ through parameter sharing. Sparsification removes redundant activations during training, and symbolification fits each surviving spline to a small library of elementary functions with affine rescaling. The paper reports that the mixed basis reaches R² = 0.999 on the rubber benchmark and transfers to the general biaxial dataset, that principal-stretch CKANs best capture tension-compression asymmetry in brain tissue, and that feature-augmented CKANs interpolate and extrapolate Shore hardness in silicone. The authors argue this shows hybrid models can be simultaneously accurate, interpretable, and extrapolative.

Load-bearing premise

The stretch-based variant of the model can only represent strain energies that are sums of one-dimensional functions of the individual stretches; if a material's response depends on how stretches interact, this architecture cannot see it.

Editorial extensions

If this is right

  • The same pipeline can deliver both a fitted constitutive law and a symbolic equation, so downstream finite-element codes can use the closed form rather than the network.
  • Functional-basis choice is decisive: for multi-axial rubber, principal-stretch or mixed bases generalize far better than principal-invariant bases, which the paper ties to earlier evidence that the first two invariants alone under-describe vulcanized rubber.
  • Feature augmentation (e.g., Shore hardness) turns a CKAN into a predictor for unseen material variants, demonstrated by leave-one-out hardness prediction.
  • Sparsification plus monotonicity yields compact, physically sensible forms: the stretch-based CKAN recovers a classic power-law rubber energy, and invariant-based brain CKANs reduce to functions of the second invariant alone.
  • The framework is not limited to rubber: brain tissue data show it handles tension-compression asymmetry with sparse data, given the right functional basis.

Reading between the lines

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

  • The paper's own results suggest the strongest test of CKAN generality is a material whose true energy is not additively separable in stretches; for such materials the stretch-based variant would need a richer ansatz, and the mixed basis would carry the burden.
  • A natural next experiment is to let an attribution-based pruning score decide which activation to keep; the paper notes that pruning by magnitude alone can discard functions that matter for stress derivatives, so better attribution could improve sparsity without accuracy loss.
  • If symbolification can be made fully automatic—the paper flags manual hyperparameter and regression tuning as a limitation—CKANs become a drop-in tool for constitutive model discovery across soft materials, potentially replacing bespoke symbolic-regression pipelines.
  • The recovered Shore-hardness law, a hyperbolic cosine in the first invariant with a quadratic hardness term, suggests a testable constitutive ansatz for silicone elastomers that experimentalists could verify with independent data.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript introduces Constitutive Kolmogorov–Arnold Networks (CKANs), which represent hyperelastic strain energy functions using KANs with spline activations, enforce constitutive normalization and optional monotonicity, sparsify the network during training, and convert the remaining activations into symbolic expressions. The framework is compared across four functional bases on Treloar's rubber data, with Kawabata's biaxial data as a validation set, applied descriptively to human brain tissue, and used in a leave-one-out study on Ecoflex silicones of varying Shore hardness. The central claim is that CKANs simultaneously achieve accuracy, interpretability, and extrapolability, including the ability to predict the behavior of previously unseen materials.

Significance. The paper is a solid engineering contribution. The continuum mechanics derivations in Appendices A and C are careful and internally consistent, and the systematic comparison of four functional bases on the same benchmark data is informative. The authors report the discovered symbolic strain energy functions in tabular form, which aids reproducibility and interpretability. The feature-augmentation idea, using Shore hardness as an additional input, is promising if the predictive claims can be substantiated. If the Ecoflex results are validated, the CKAN framework would be a useful addition to the hybrid constitutive modeling toolbox. However, the strongest cross-material predictive claim rests on a single leave-one-out experiment whose reported parameter table contains an unexplained duplication, and this issue must be resolved before the claim can be accepted at face value.

major comments (3)
  1. [Table 2, Section 5.3] The rows for 'Leave out Shore OO-20' and 'Leave out Shore OO-30' are numerically identical to four significant figures in all six material parameters (a=6.4998, b=0.3427, c=0.4190, d=0.0100, e=0.0800, f=-0.6707). These two leave-one-out training sets differ, and independent optimizations on different datasets would not be expected to converge to exactly the same parameter values. This situation indicates either a transcription error in the table or a model that is insensitive to which hardness level is withheld; in the latter case, the claimed predictive capability for unseen Shore hardness levels would not be supported. Please re-run the two experiments, report the actual parameter sets and optimization seeds, and correct the table if needed.
  2. [Abstract, Section 5.1, Section 6] The Kawabata validation is a same-material, different-deformation-mode generalization test: as the paper states, Treloar and Kawabata used vulcanized rubber of the same chemical composition. The phrase 'predicting previously unseen materials' in the Abstract and Conclusions is therefore supported only by the Ecoflex experiment. Please qualify the extrapolation claim so that it is clear that the Treloar-to-Kawabata result demonstrates generalization to new deformation modes of a known material, not prediction for an unseen material.
  3. [Section 2.2, Eq. (13), Appendix B] The principal stretch-based CKAN and the mixed CKAN impose structural assumptions: the Valanis–Landel additive separability for the former and an additive invariant-plus-stretch split for the latter. Not every isotropic hyperelastic strain energy function is additively separable in this way, so these CKAN variants cannot represent non-separable coupling between stretches. This restriction is acknowledged in the derivation but should be stated explicitly as a limitation in Sections 3.2 and 6, because the abstract and conclusions claim a general resolution of the accuracy/interpretability trade-off without this caveat.
minor comments (6)
  1. [Section 4.1] The heading 'Sparsificaiton' contains a typo and should read 'Sparsification'.
  2. [Figure E.3] The figure titles repeat the typo 'Princiapl invariant-based'; this should be corrected to 'Principal invariant-based'.
  3. [Appendix C.1] The sentence 'For an isotropic incompressible material, the' is incomplete and should either be finished or removed.
  4. [Data availability] The manuscript does not include a data or code availability statement. Given that the numerical results are central to the claims, please provide a statement and, if possible, release code and data to reproduce the results, especially the Ecoflex leave-one-out study.
  5. [Figure 9a] The reported 'MSE = 0.0000' to four decimal places is not informative; please report the value with more significant digits or in scientific notation.
  6. [Section 5.3] The leave-one-out study uses only four hardness levels, and the reported R² and MSE values are single point estimates. Please state whether repeated training runs with different random seeds were performed and report the resulting variability, or at least note the absence of such a stability check more prominently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the CKAN's symbolic laws are fitted representations validated against external data, not reductions to their training inputs.

full rationale

The derivation chain is self-contained: CKAN's strain energy ΨCKAN (Eq. 21) is a fitting architecture whose spline parameters are optimized against stress data, and the symbolification step (Sec. 4.3) is an explicit post-processing compression of the fitted splines rather than a parameter-free derivation; the paper does not represent it differently. The central extrapolation claims are tested against data not used in fitting: Section 5.1 trains only on Treloar's uniaxial, equibiaxial, and pure shear data and validates on Kawabata's general biaxial set, and Section 5.3's leave-one-out Ecoflex analysis withholds entire Shore hardness levels, so the held-out curves are out-of-sample. No equation defines the target in terms of the quantity said to predict it; normalization terms Ψσ and Ψε are constraints, not predictions. Self-citations to CANN work appear only as background and conceptual lineage, while the load-bearing evidence is in-paper external benchmarks. One non-circular reproducibility concern is that Table 2 lists identical parameter tuples for the OO-20 and OO-30 leave-one-out folds although those folds have different training sets; this weakens the Ecoflex predictive demonstration but is an inconsistency, not a reduction of prediction to fit by construction.

Assumptions & free parameters 5 free parameters · 9 assumptions · 0 invented entities

The central claim rests on standard continuum mechanics assumptions and on architectural choices that restrict the class of representable strain energy functions. The discovered symbolic expressions' coefficients are fitted to data, and several hyperparameters (network topology, regularization strength, spline grid, symbolic library) are selected by hand without reported values.

free parameters (5)
  • Material parameters of discovered symbolic models = e.g., Treloar mixed basis: a=2.8333 MPa, b=7.6034e-4, c=4.9633e-3 MPa, d=0.0315, e=2.3312 MPa, f=0.0219
    These coefficients are optimized against stress data after symbolification; they are part of the fitted model output.
  • Regularization strength Λ = not reported
    Hyperparameter in Eq. (27) controlling sparsification; unspecified numerically.
  • Spline grid size and B-spline order = not reported
    KAN 1.0 default settings are used, but no grid size or spline order is given.
  • Symbolic function library B = elementary functions (monomials, ln, exp, cos, etc.)
    User-chosen library constrains the symbolic search space.
  • Network topologies N = e.g., [3,4,1], [3,1,1]
    Chosen per dataset; affects model capacity.
assumptions (9)
  • standard math Kolmogorov-Arnold representation theorem: any continuous multivariate function on [0,1]^n can be represented as a finite composition of univariate functions and sums
    Invoked in Section 3.1 to justify KANs as universal approximators.
  • domain assumption Existence of a strain energy function Ψ(F) with P = ∂Ψ/∂F (hyperelasticity)
    Fundamental assumption of hyperelasticity, Section 2.2, Eq. (6).
  • domain assumption Objectivity is ensured by expressing Ψ as a function of C = F^T F only
    Section 2.2, Eq. (7).
  • domain assumption Isotropic materials: Ψ is a scalar isotropic tensor function, expressible in functional bases of invariants or stretches
    Section 2.2, Eq. (9).
  • domain assumption Incompressibility (J=1) for rubber and brain tissue
    Used in all examples, Section 5; determines the third stretch and introduces hydrostatic pressure p.
  • domain assumption Additive separability (Valanis-Landel hypothesis) for principal stretch-based CKANs: Ψ = Σ ω1(λα) + Σ ω−1(λα−1)
    Section 3.2, Eq. (13) and Appendix B; this restricts the function class and is not universal.
  • domain assumption Plane stress assumption to determine the hydrostatic pressure p in biaxial and shear tests
    Appendix A.2, e.g., P33=0 in Eqs. (A.20), (A.23), (A.26), (A.31).
  • domain assumption Monotonicity of strain energy with respect to strain inputs (and Shore hardness for Ecoflex)
    Section 5.3 and Appendix D.2; imposed via MonoKAN constraints, physically motivated but a priori restriction.
  • domain assumption Treloar and Kawabata vulcanized rubbers have the same chemical composition, permitting cross-dataset validation
    Section 5.1, citing [56]; if false, the generalization test is weakened.

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Cite this review

Pith. "Pith review of Constitutive Kolmogorov-Arnold Networks (CKANs): Combining Accuracy and Interpretability in Data-Driven Material Modeling." pith.science (2026). https://pith.science/paper/RVR4HJQ6

@misc{pith2026250205682,
  author       = {Pith},
  title        = {Pith review of: Constitutive Kolmogorov-Arnold Networks (CKANs): Combining Accuracy and Interpretability in Data-Driven Material Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RVR4HJQ6}},
  note         = {Machine review of arXiv:2502.05682}
}
read the original abstract

Hybrid constitutive modeling integrates two complementary approaches for describing and predicting a material's mechanical behavior: purely data-driven black-box methods and physically constrained, theory-based models. While black-box methods offer high accuracy, they often lack interpretability and extrapolability. Conversely, physics-based models provide theoretical insight and generalizability but may not capture complex behaviors with the same accuracy. Traditionally, hybrid modeling has required a trade-off between these aspects. In this paper, we show how recent advances in symbolic machine learning, specifically Kolmogorov-Arnold Networks (KANs), help to overcome this limitation. We introduce Constitutive Kolmogorov-Arnold Networks (CKANs) as a new class of hybrid constitutive models. By incorporating a post-processing symbolification step, CKANs combine the predictive accuracy of data-driven models with the interpretability and extrapolation capabilities of symbolic expressions, bridging the gap between machine learning and physical modeling.

Figures

Figures reproduced from arXiv: 2502.05682 by the authors.

Figure 1
Figure 1. Constitutive Kolmogorov-Arnold Networks (CKANs). Given experimental stretch λ and stress P data and po￾tentially also non-mechanical data f (such as microstructural imaging data or information on materials processing), a CKAN learns the strain energy function Ψ of a hyperelastic material. It inherently satisfies essential mathematical or physical requirements, such as objectivity, symmetry properties, or thermodynam… view at source ↗
Figure 2
Figure 2. Representation of special planar modes of deformation of a thin incompressible membrane using three different functional bases: the complexity of the functional forms required to represent uniaxial tension, equibiaxial tension, and pure or simple shear vastly differs depending on whether one relies on representing them in terms of (a) principal stretches {λ1, λ2}, (b) principal invariants {I1, I2}, (c) modified inva… view at source ↗
Figure 3
Figure 3. Comparison of a Kolmogorov–Arnold Network (KAN) and a feedforward neural network (FFNN). A KAN (a) can be interpreted as a generalization of an FFNN (b) where the weights are no longer constants but nonlinear functions, typically spline functions of higher order, and where as a consequence the intra-neural (typically nonlinear) activation function σ is no longer needed. While in an FFNN the weights wl,j,i are adjust… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Schematic overview of the Constitutive Kolmogorov–Arnold Network (CKAN) framework: The right Cauchy Green deformation tensor C and the feature vector f serve as input to the CKAN. Depending on the chosen functional basis F (principal stretches, principal invariants, mo…
Figure 5
Figure 5. Figure 5: Examples of Kolmogorov–Arnold Networks (KANs) representing the strain energy function ΨKAN, where the functional basis F = {I1, I2, I3} was chosen and reduced to the setting of an incompressible material so that I3 is no longer required as an explicit part of the input…
Figure 6
Figure 6. Figure 6: Comparison of the descriptive performance on Treloar’s data: Mean squared error (MSE) and coefficient of determination (R2 ) of each functional basis for predicting the stress component P11. For simplicity, we report in this section and the related appendices only Ψˆ K…
Figure 7
Figure 7. Figure 7: Comparison of the generalization performance on Kawabata’s data: Mean normalized mean squared error (MNMSE) of each functional basis for predicting the stress components P11 and P22 in the small and large stretch regimes of Kawabata’s data. shows its fitting results on…
Figure 8
Figure 8. Figure 8: Training and validation results for vulcanized rubber. (a) Fitting performance on Treloar’s data [51]. (b) Architecture of the final CKAN after training; unconnected nodes indicate pruned activation functions of the network. (c)– (f) Generalization performance of the m…
Figure 9
Figure 9. Figure 9: Descriptive performance of CKAN based on principal stretches for human brain (cortex) tissue [59]: (a) Performance of CKAN simultaneously fitted to uniaxial tension, compression, and simple shear data. (b) Resulting strain energy functions and material parameters. (c) …
Figure 10
Figure 10. Figure 10: Comparison of the descriptive performance on the brain tissue data. Mean squared error (MSE) and coefficient of determination (R2 ) of the symbolic strain energy functions for the human brain (cortex) each using a different functional basis. highlighting the pivotal r…
Figure 11
Figure 11. Figure 11: Comparison of the predictive performance on the Ecoflex silicone polymer data. Mean squared error (MSE) and coefficient of determination (R2 ) of the stress curves generated from the discovered symbolic principal invariant-based strain energy functions for Ecoflex sil…
Figure 12
Figure 12. Figure 12: Training results (descriptive performance) and validation results (predictive performance) for Ecoflex silicone polymer data from [54] for principal invariant-based symbolic CKANs applied in leave-one-out cross-validation scheme. The lines represent the model response…
Figure 13
Figure 13. Figure 13: CKAN architectures obtained during the leave-one-out cross-validation test for Ecoflex silicone polymer data. The red graphs illustrate the symbolic activation functions fi,j,k, and the red lines are the only remaining connections after the training. Unconnected nodes…

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Pith tools

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