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Learning constitutive relations from experiments: 1. PDE constrained optimization

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Constitutive behavior can be identified from full-field experiments by solving a PDE-constrained inverse problem whose adjoint gradient costs about one extra forward solve per iteration, nearly independent of the number of material…

desk verdict A competent adjoint-based parameter identification paper with an honest treatment of non-uniqueness, but the validation is too clean to support the 'learning constitutive relations' framing. read the letter →

arxiv 2412.02864 v1 pith:Y4SKFWYW submitted 2024-12-03 cond-mat.mtrl-sci physics.comp-ph

Andrew Akerson , Aakila Rajan , Kaushik Bhattacharya This is my paper · ORCID
classification cond-mat.mtrl-sciphysics.comp-ph MSC 74C1074S0565N21
keywords constitutiverelationidentificationPDE-constrainedoptimizationadjointmethodelasto-viscoplasticityfull-fieldmeasurementsinverseproblemsdigitalimagecorrelationinternalvariables
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

Learning a material's constitutive relation is an indirect inverse problem: measure displacements and forces from an experiment, then find the stress-strain law that makes a simulation reproduce those observations. This paper shows that the problem can be posed as a PDE-constrained optimization, with the balance laws as constraints, and that the adjoint method supplies the parameter gradient at the cost of roughly one extra forward solve. The central scaling claim is Eq. (11): each optimization iteration costs A + B*NP with A >> B, so the cost barely grows with the number of parameters, unlike gradient-free or finite-difference approaches. The method is demonstrated on synthetic elasto-viscoplastic data from a quasistatic plate-with-hole compression and a dynamic impact test; the recovered parameters reproduce the observations and pass an independent uniaxial stress-strain test, even though some individual hardening parameters differ from the generating values. If this holds, constitutive relations could be identified from complex, heterogeneous experiments rather than idealized uniform tests, and the approach extends to hyperparameterized models such as neural networks.

What carries the argument

The machinery is the adjoint of the initial-boundary value problem that governs the experiment. The objective is augmented with the weak form of the balance laws and internal-variable evolution (Eq. 5), and the adjoint states are chosen to satisfy a backward-in-time system (Eqs. 10 and 18) that removes all terms containing unknown solution sensitivities. This leaves the parameter gradient (Eqs. 9 and 17) as a single quadrature, so each gradient evaluation costs one forward solve plus one adjoint solve, independent of the parameter count. The parameter update is handled by the Method of Moving Asymptotes, and the forward-adjoint pair is discretized with P1 finite elements in space with quadrature-point plastic variables.

What would settle it

Run the identical inversion on data generated by a materially different constitutive law, such as a different hardening form or a model with damage or softening, or on real experimental data with known measurement noise, and then test the recovered model on an independent non-proportional loading path not used in the objective. A concrete version: add roughly one percent noise to the synthetic displacements and reaction forces in the quasistatic plate-with-hole test and check whether the zero-shot uniaxial error stays below a few percent.

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Extended reading notes

Core claim

The paper claims that full-field experimental observations, combined with the balance laws, carry enough information to identify the constitutive behavior of an inelastic material through a PDE-constrained optimization. The forward problem is the initial-boundary value problem of the experiment; the objective compares computed and measured displacements and reaction forces; and the adjoint method yields the parameter gradient without forming solution sensitivities. For a general Perzyna-type internal variable theory, the adjoint system (10) is linear in space and quasilinear in time, solved backward in time, and for J2 elasto-viscoplasticity with power-law hardening the sensitivity reduces to (17). Two synthetic demonstrations, one quasistatic and one dynamic, drive the objective down by several orders of magnitude from a poor initial guess, show insensitivity to initial guess, objective weights, and mesh resolution, and yield parameters that match the yield strength while differing in hardening parameters yet agreeing on independent uniaxial response. The paper interprets this as observational degeneracy: the method identifies models that reproduce the measured response, not necessarily the underlying parameter values.

Load-bearing premise

The load-bearing premise is that the true material's response lies inside the chosen parametric family of Perzyna-type internal-variable models, since the inversion only tunes those parameters and the noise-free synthetic data come from the very same forward solver used in the inversion.

Editorial extensions

If this is right

  • Material characterization no longer requires specimens that achieve uniform stress and strain; a complex geometry that generates many strain paths can substitute for many separate idealized tests.
  • Because per-iteration cost is nearly independent of the number of parameters, constitutive models with hundreds or thousands of parameters, including neural networks, become practical targets for identification from experiments.
  • The formulation applies to history-dependent, rate-dependent inelastic response with a yield surface, a setting where numerical differentiation of the objective would be fragile or prohibitive.
  • A single inversion can combine data from multiple tests, such as different strain rates or geometries, in one objective; in the quasistatic example a single final displacement snapshot plus a force history sufficed to fix the observed response.
  • Two materially different parameter sets can fit the same observations, so the meaningful output of the method is a model that reproduces measured and independently tested response, not a unique parameter vector.

Reading between the lines

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

  • If the per-iteration cost really is dominated by the forward and adjoint solves, then the practical bottleneck for neural-network parameterizations will be backpropagating through the adjoint update rather than the raw number of network weights; the announced part 3 of this series is where that claim gets tested.
  • The validation uses noise-free synthetic data generated by the same forward solver used in the inversion, so real-world applicability hinges on tolerance to measurement noise and model-form error; a natural companion study would contaminate synthetic data with DIC-like noise and test an out-of-family material law.
  • The quasistatic result that the final snapshot plus force history is enough suggests specimen geometry can be engineered to maximize information content, for example asymmetric holes or multi-rate loading paths, to shorten experiments or reduce the number of tests needed.
  • The observed non-uniqueness implies that reporting a single best-fit parameter vector will understate uncertainty; prediction-oriented validation, such as the zero-shot uniaxial test used here, is the correct criterion for whether an identified law is useful.
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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 / 5 minor

Summary. The paper proposes a PDE-constrained optimization framework for identifying constitutive parameters of history-dependent materials from full-field displacement measurements and macroscopic force histories. The forward problem is a boundary value problem for a J2 elasto-viscoplastic material with power-law hardening and rate dependence; the sensitivity of the objective is computed with an adjoint method, linear in space and quasilinear in time, and the parameters are updated with a gradient-based optimizer. The method is demonstrated on two synthetic benchmarks: a quasistatic compression of a plate with a hole using only the final displacement snapshot and force history, and a dynamic compression of a thin annular specimen reduced to a two-dimensional problem. The authors report that the recovered parameters differ from the generating parameters even though the training objective is very small, and they introduce a zero-shot uniaxial stress-strain test to show predictive equivalence. They conclude that the method is accurate and efficient, scales nearly independently of the number of parameters, and is suited to hyperparameterized constitutive models.

Significance. If the central claims are established, this would be a valuable contribution to full-field inverse material characterization: a systematic adjoint formulation for a non-smooth, history-dependent constitutive model, with explicit numerical schemes in the appendices and careful robustness studies over initial guesses, objective weights, mesh size, and specimen geometry. The authors deserve credit for acknowledging the degeneracy of the parameter-to-observation map and for testing predictive equivalence with an independent uniaxial objective. The scalability claim, if supported, would motivate extensions to neural-network constitutive models. However, the reported parameter non-uniqueness and the absence of noise and model-form validation mean the paper currently demonstrates a useful parameter-fitting framework rather than the stronger claim of learning constitutive relations from experiments.

major comments (3)
  1. [§3, Tables 1–3; §3.1 concluding sentence] The claim that the method 'accurately recovers elasto-viscoplastic material parameters' is not supported by the reported results. In the quasistatic example the recovered εp0 is 0.0393 versus the generating value 0.0225 (about 75% error) while the objective is 5.59×10⁻⁹; in the dynamic example the recovered m is 4.50 versus 2.00 and εp0 is 0.0286 versus 0.0225 while the objective is 1.44×10⁻⁴. Table 3 shows that different initial guesses converge to materially different parameter vectors (m from 1.09 to 4.50, εp0 from 0.0277 to 0.0781) with final objectives spanning 0.87×10⁻⁴ to 8.57×10⁻⁴. The independent uniaxial test, Eq. (21), gives Oind = 0.0069 for the quasistatic case and 0.021 for the dynamic case and probes only a single loading path. Thus the paper has not established that the optimized parameters identify the constitutive relation; it has shown that the calibration data admit an equivalence class of parameters. The authors acknowledge this degeneracy, but the central claim of learning constitutive relations requires either an identifiability/observability analysis of the parameter-to-observation map or a multi-path, multi-axial hold-out validation demonstrating that the recovered class members predict equivalent responses outside the calibration set.
  2. [§2.1.4, Eq. (11)] The cost model Cost(NP) = A + B NP with A >> B is asserted rather than demonstrated. The discussion omits the handling of the forward trajectory during the backward adjoint solve: the adjoint equations use forward fields at each time step, and neither a storage nor a recomputation strategy is described. All demonstrations use only five parameters, so the claimed near-independence of the number of parameters—which is the paper's motivation for neural-network extensions—is not verified. I recommend adding a scaling experiment with larger NP (for example using a smooth surrogate problem) or providing a more precise analysis of the constants in Eq. (11) including trajectory management and the cost of the parameter update.
  3. [§3.1 and §3.2 validation setup] The synthetic data are generated by the same forward solver and the same material model used in the inversion, with no noise, and the dynamic experiment is reduced by the uniform-axial-strain assumption (Eq. (23)). This setup cannot reveal whether the method tolerates measurement noise, DIC interpolation error, or model-form error, all of which are central to accurate identification from experiments. The dynamic test is self-validating in that both data generation and inversion use Eq. (51). The authors should either add a noise robustness study (for example, perturbing uexp and f_R at levels representative of DIC) and a model-form-error test (for example, generating data with a slightly different hardening law), or explicitly scope the claim to noise-free, in-family synthetic benchmarks.
minor comments (5)
  1. [§3.2, Sensitivity to initial guess] The text states that the objective remains on the order of 10⁻⁹ after the initial-guess sensitivity study, but Table 3 reports final objectives around 10⁻⁴; the text and table should be made consistent.
  2. [Eq. (18)] Equation (18) contains a typographical artifact: '= − ∂o/∂u · on Ω' should read '= −∂o/∂u on Ω'.
  3. [Figure 3 caption] The caption of Figure 3(f) says 'the rsults of the independent stress-strain test'; this should be corrected to 'results'.
  4. [Eq. (21)] The definition of Oind integrates over strain, but the strain interval is not specified; adding the integration range would make the reported numbers reproducible.
  5. [§2.2.2 and Appendix A] The adjoint formulation for elasto-viscoplasticity is presented for the smooth flow rule, but the treatment of the elastic unloading/yield boundary is not discussed; a sentence describing how the active plastic region is handled in the adjoint evolution would improve the reproducibility of the method.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the optimization derivation is self-contained, and the acknowledged parameter degeneracy is an identifiability limitation rather than a circular step.

full rationale

The paper's derivation chain is not circular. The forward problem (Eq. 2) and objective (Eq. 4) are defined independently, the adjoint sensitivity (Eq. 9) is derived by differentiating the objective subject to the constraints, and the parameters are updated by gradient-based optimization. Nothing in this chain defines the constitutive parameters in terms of the quantities they are supposed to predict. The recovered parameters are then evaluated on an independent zero-shot uniaxial test (Eq. 21) that was not part of the calibration objective, so the validation is not forced by construction. The main caveat is that the synthetic data are generated by the same forward solver used in the inversion, which limits the benchmark to in-family, noise-free data; however, this is a testing limitation, not circular reasoning. The paper explicitly acknowledges the non-uniqueness of the recovered parameters in Section 3.1: 'the experiment with three tests at three strain rates are unable to distinguish between the two sets of parameters' and later 'reflects the degeneracy of the constitutive model.' This admission is the opposite of a circular claim: the authors do not assert uniqueness, and they supplement the calibration objective with an independent test. Self-citations in the references (e.g., [8], [31], [33], [60]) are contextual and are not load-bearing for the derivation. The cost-scaling claim in Eq. (11) follows structurally from the algorithm's decomposition into forward solve, adjoint solve, and parameter update, and is not a fitted prediction. Overall, no quoted step reduces to its own inputs, so the circularity score is minimal.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the assumed parametric model class and on synthetic validation; no new physical entities are introduced.

free parameters (2)
  • objective weights alpha_u, alpha_f = alpha_f/alpha_u in {0.1, 1, 10} for sensitivity tests; main runs use alpha_f/alpha_u = 1
    Scaling factors balancing displacement and force terms in the objective (Eqs. 20, 24); chosen by hand and tested only over a narrow range.
  • number of optimization iterations = 300 for quasistatic; convergence-based for dynamic
    Arbitrary stopping criterion; results may shift with more iterations, though objective already small.
assumptions (6)
  • domain assumption Balance laws (momentum, mass) are the only constraints needed to close the system.
    Standard continuum mechanics foundation invoked throughout Section 2.
  • domain assumption The true material follows a Perzyna-type internal variable constitutive relation with a known parametric form for S and R.
    The inversion searches only within this parametric family; see Eqs. (1) and Section 2.2.1.
  • domain assumption Small strain assumption for the J2 model.
    Section 2.2.1 uses linearized strain epsilon = (grad u + grad u^T)/2, restricting results to small deformations.
  • ad hoc to paper Synthetic data is generated with the same forward solver and model used in inversion, with no noise.
    Section 3: validation uses simulated ground truth, avoiding measurement error and model-form error, which may not transfer to real experiments.
  • ad hoc to paper Uniform axial strain assumption for the dynamic annular specimen.
    Appendix C reduces the 3D problem to 2D via Eq. (23), limiting generality to thin specimens.
  • standard math Differentiability and regularity conditions for the adjoint derivation (integration by parts, terminal conditions).
    Standard adjoint calculus assumptions, used in Section 2.1.3 and Appendix A.

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Cite this review

Pith. "Pith review of Learning constitutive relations from experiments: 1. PDE constrained optimization." pith.science (2026). https://pith.science/paper/Y4SKFWYW

@misc{pith2026241202864,
  author       = {Pith},
  title        = {Pith review of: Learning constitutive relations from experiments: 1. PDE constrained optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4SKFWYW}},
  note         = {Machine review of arXiv:2412.02864}
}
read the original abstract

We propose a method to accurately and efficiently identify the constitutive behavior of complex materials through full-field observations. We formulate the problem of inferring constitutive relations from experiments as an indirect inverse problem that is constrained by the balance laws. Specifically, we seek to find a constitutive behavior that minimizes the difference between the experimental observation and the corresponding quantities computed with the model, while enforcing the balance laws. We formulate the forward problem as a boundary value problem corresponding to the experiment, and compute the sensitivity of the objective with respect to model using the adjoint method. The resulting method is robust and can be applied to constitutive models with arbitrary complexity. We focus on elasto-viscoplasticity, but the approach can be extended to other settings. In this part one, we formulate the method and demonstrate it using synthetic data on two problems, one quasistatic and the other dynamic.

Figures

Figures reproduced from arXiv: 2412.02864 by the authors.

Figure 1
Figure 1. Schematic representation of the iterative algorithm to obtain to find material parameters. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Results for the quasistatic compression of a thick plate with offset hole. (a) Geometry [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Results for the dynamic compression of circular annulus. (a) Geometry and deformation. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Objective curves with respect to iterations for different mesh sizes. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Results for the dynamic compression of elliptic annulus . (a) Normalized objective from [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Results for the dynamic compression of flower-shaped specimen . (a) Normalized objective [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Results for the dynamic compression of disc with flower-shaped hole . (a) Normalized [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.