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arxiv: 2603.12365 · v2 · submitted 2026-03-12 · ❄️ cond-mat.mtrl-sci · cs.LG· cs.NA· math.NA· physics.comp-ph· stat.CO

Optimal Experimental Design for Reliable Learning of History-Dependent Constitutive Laws

Kaushik Bhattacharya , Lianghao Cao , Andrew Stuart This is my paper

Pith reviewed 2026-05-15 11:22 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cs.LGcs.NAmath.NAphysics.comp-phstat.CO
keywords optimal experimental designconstitutive modelshistory-dependentBayesian inferenceviscoelastic solidsparameter identifiabilityinformation gainexperimental design optimization
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The pith

A Bayesian optimal experimental design framework uses expected information gain to optimize specimen geometries and loading paths for reliable identification of parameters in history-dependent constitutive models.

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

History-dependent constitutive models act as macroscopic closures for micromechanical effects, yet their parameters are often poorly constrained by data collected under a fixed experimental budget. The paper develops a Bayesian design framework that selects geometries and loading sequences to maximize the expected reduction in uncertainty about those parameters. Two practical approximations—a Gaussian form for the information gain and a surrogate for the Fisher information matrix—make the optimization tractable even when forward simulations are expensive and data are high-dimensional. Numerical experiments on uniaxial viscoelastic tests demonstrate that the resulting designs produce image and force measurements that tighten posterior distributions more effectively than random choices, with the largest gains appearing for parameters that encode memory.

Core claim

The central claim is that design utility can be quantified as expected information gain and maximized, via a Gaussian approximation and a Fisher-matrix surrogate, to select specimen shapes and loading histories that yield data significantly more informative about history-dependent parameters than conventional choices; this is verified on viscoelastic solids where memory-effect parameters exhibit the clearest improvement in identifiability.

What carries the argument

Bayesian optimal experimental design that maximizes Gaussian-approximated expected information gain, supported by a surrogate Fisher information matrix for batched optimization.

If this is right

  • Optimized geometries and paths produce image and force data that reduce posterior variance on memory-effect parameters more than random designs do.
  • The Gaussian approximation enables rapid in silico ranking of candidate experiments without repeated full posterior sampling.
  • The Fisher-matrix surrogate amortizes the cost of utility evaluation, permitting joint optimization over batches of tests.
  • The same workflow applies to any history-dependent constitutive law once a forward simulator is available.

Where Pith is reading between the lines

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

  • The framework could be applied to other history-dependent phenomena such as rate-dependent plasticity or damage evolution by substituting the appropriate forward model.
  • Physical validation would consist of fabricating the optimized specimen geometries, collecting the predicted data, and checking whether the resulting posterior widths match the in silico predictions.
  • When the posterior is known to be multimodal, replacing the Gaussian approximation with a more flexible utility estimator would be a direct next step.
  • The approach suggests that experimental budgets for material characterization can be reallocated from repeated random trials to fewer, higher-value tests.

Load-bearing premise

The Gaussian approximation to expected information gain remains sufficiently accurate for design optimization even when the true posterior over constitutive parameters is non-Gaussian or multimodal.

What would settle it

A Monte Carlo estimate of true expected information gain for the candidate designs that differs substantially from the Gaussian prediction would indicate that the reported optimum is not maximal.

Figures

Figures reproduced from arXiv: 2603.12365 by Andrew Stuart, Kaushik Bhattacharya, Lianghao Cao.

Figure 1
Figure 1. Figure 1: A visualization of a model for the described uniaxial test with loading path and specimen shape [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A visualization of the observation operator for a single snapshot of image data at time [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example of the uniaxial testing setup and the design variables. ( [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An example of simulated force and image data for learning linear viscoelasticity. This simulated [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The surrogate FIM testing accuracy as a function of training sample size for designing uniaxial [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The optimized designs of uniaxial test for reliable learning of linear viscoelasticity found by EGA [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visualizations and statistics of the 95% CIs for constitutive parameters of linear viscoelasticity [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The normalized EGA design utility for learning marginal parameters of linear viscoelasticity [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Relation between principal-stress-based stress-state entropy and EGA for randomly selected de [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The optimal batched designs of uniaxial testing for learning linear viscoelasticity found by ESFIM [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Visualization and statistics of the EGA design utility values of batched designs for uniaxial [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: An example of simulated force and image data for learning nonlinear viscoelasticity. This [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Visualization and analysis of ESFIM-optimized designs for uniaxial testing of nonlinear viscoelas [PITH_FULL_IMAGE:figures/full_fig_p038_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The experimental designs with batch sizes 1–3 found via ESFIM maximization for uniaxial [PITH_FULL_IMAGE:figures/full_fig_p039_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Visualizations and statistics of the 95% CIs for constitutive parameters of nonlinear viscoelasticity inferred from experiments with random and ESFIM-optimized designs. We consider design optimization without (λ = 0) and with (λ = 0.25) strain-rate regularization. (Top) The box plots of the 95% CI sizes. The distributions arise from the prior-based variation in the experimental outcomes. For random design… view at source ↗
Figure 16
Figure 16. Figure 16: The normalized EGA design utility for learning marginal parameters of nonlinear viscoelasticity [PITH_FULL_IMAGE:figures/full_fig_p041_16.png] view at source ↗
read the original abstract

History-dependent constitutive models serve as macroscopic closures for the aggregated effects of micromechanics. Their parameters are typically learned from experimental data. With a limited experimental budget, eliciting the full range of responses needed to characterize the constitutive relation can be difficult. As a result, the data can be well explained by a range of parameter choices, leading to parameter estimates that are uncertain or unreliable. To address this issue, we propose a Bayesian optimal experimental design framework to quantify, interpret, and maximize the utility of experimental designs for reliable learning of history-dependent constitutive models. In this framework, the design utility is defined as the expected reduction in parametric uncertainty or the expected information gain. This enables in silico design optimization using simulated data and reduces the cost of physical experiments for reliable parameter identification. We introduce two approximations that make this framework practical for advanced material testing with expensive forward models and high-dimensional data: (i) a Gaussian approximation of the expected information gain, and (ii) a surrogate approximation of the Fisher information matrix. The former enables efficient design optimization and interpretation, while the latter extends this approach to batched design optimization by amortizing the cost of repeated utility evaluations. Our numerical studies of uniaxial tests for viscoelastic solids show that optimized specimen geometries and loading paths yield image and force data that significantly improve parameter identifiability relative to random designs, especially for parameters associated with memory effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes a Bayesian optimal experimental design framework for learning parameters of history-dependent constitutive models. Design utility is defined as expected information gain (EIG); two approximations (Gaussian EIG and surrogate Fisher information) are introduced to enable tractable optimization with expensive forward models and high-dimensional data. Numerical studies on uniaxial tests for viscoelastic solids claim that optimized specimen geometries and loading paths yield image/force data that significantly improve parameter identifiability relative to random designs, especially for memory-effect parameters.

Significance. If the approximations are shown to be reliable, the work offers a practical route to designing experiments that maximize information for complex, history-dependent material models, potentially lowering experimental costs while improving parameter reliability. The emphasis on memory parameters and the use of in-silico optimization are timely contributions to materials characterization.

major comments (3)
  1. [Numerical studies] Numerical studies section: the claim of 'significant' improvement in parameter identifiability is not supported by any reported quantitative metrics (specific EIG values, posterior variance reductions, or error bars on the gains versus random designs), making it impossible to judge the practical magnitude of the benefit.
  2. [Approximations section] Gaussian approximation to EIG (introduced for design optimization and utility evaluation): no validation or accuracy assessment is provided against exact mutual information or for cases where the posterior over history-dependent parameters is non-Gaussian or multimodal, which is a load-bearing assumption given the nonlinear forward map from parameters to time-series data.
  3. [Surrogate approximation] Surrogate Fisher-information approximation for batched designs: the manuscript does not quantify the additional error this introduces relative to direct EIG evaluation, nor does it test whether the surrogate preserves correct ranking of designs when the Gaussian assumption is already approximate.
minor comments (1)
  1. [Introduction] Notation for the forward model and likelihood should be introduced earlier and used consistently when describing the EIG integral.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and describe the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Numerical studies] Numerical studies section: the claim of 'significant' improvement in parameter identifiability is not supported by any reported quantitative metrics (specific EIG values, posterior variance reductions, or error bars on the gains versus random designs), making it impossible to judge the practical magnitude of the benefit.

    Authors: We agree that quantitative metrics are required to support the claims. In the revised manuscript we will report explicit EIG values for optimized versus random designs, posterior variance reductions for each parameter (including memory-effect parameters), and standard deviations computed over repeated random-design realizations. revision: yes

  2. Referee: [Approximations section] Gaussian approximation to EIG (introduced for design optimization and utility evaluation): no validation or accuracy assessment is provided against exact mutual information or for cases where the posterior over history-dependent parameters is non-Gaussian or multimodal, which is a load-bearing assumption given the nonlinear forward map from parameters to time-series data.

    Authors: We acknowledge the absence of validation. The revision will add a dedicated subsection that compares the Gaussian EIG approximation against Monte-Carlo estimates of exact mutual information on representative designs and examines the empirical posterior distributions to assess deviations from Gaussianity. revision: yes

  3. Referee: [Surrogate approximation] Surrogate Fisher-information approximation for batched designs: the manuscript does not quantify the additional error this introduces relative to direct EIG evaluation, nor does it test whether the surrogate preserves correct ranking of designs when the Gaussian assumption is already approximate.

    Authors: We agree that error quantification and ranking preservation must be demonstrated. The revised manuscript will include direct comparisons of the surrogate Fisher-information utility against full EIG evaluations on a validation set of designs and will report rank-correlation statistics to confirm that design ordering is preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper explicitly defines design utility as expected information gain (EIG) and introduces the Gaussian approximation and Fisher-information surrogate as practical computational tools for optimization with expensive models. The central numerical claim—that optimized geometries and paths improve identifiability relative to random designs—is demonstrated via forward simulations and does not reduce by construction to any fitted parameter, self-citation, or renamed input. No load-bearing step equates a prediction to its own definition or prior author result; the framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard Bayesian assumptions that the forward model is simulatable and that information gain is a suitable utility; no free parameters or invented entities are introduced in the abstract description.

axioms (1)
  • domain assumption The constitutive model parameters can be treated as random variables with a prior, and the forward simulation produces observable data (force and images).
    Required for the expected information gain to be well-defined.

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