cond-mat.mtrl-sciMar 12, 2026

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

Authors: Kaushik Bhattacharya, Lianghao Cao, Andrew Stuart

Organizations: California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA.

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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.

Figures & tables

Figure 1: A visualization of a model for the described uniaxial test with loading path and specimen shape design. The dots on the loading path, i.e., the prescribed strain, are the control points. The color on the deformed material represents the magnitude of the displacement.
Figure 2: A visualization of the observation operator for a single snapshot of image data at time tt for the uniaxial testing described in Section 3.1 .
Figure 3: An example of the uniaxial testing setup and the design variables. ( Left ) The setup and the specimen geometry design. ( Right ) The loading path design.
Physical ParameterDescriptionBound
log⁡E1(0)\log E_{1}^{(0)}Log. of equilibrium modulus in 11 -dir.(2.0, 3.5)
rEr_{E}Equilibrium modulus anisotropy, E2/E1E_{2}/E_{1}(0.2, 1)
rGr_{G}Normalized shear, G/E1E2G/\sqrt{E_{1}E_{2}}(0.2, 0.8)
rνr_{\nu}Poisson’s ratio product, ν12ν21\nu_{12}\nu_{21}(0.0, 0.2)
αc\alpha_{c}Material axis angle(- π\pi /4, π\pi /4)
fif_{i} , i=1,2i=1,2Total relaxed fraction in dir. ii(0.2, 0.8)
Table 1: The bounds for the uniform distributions of the physical parameters in the re-parameterized linear viscoelastic constitutive model.
Figure 4: An example of simulated force and image data for learning linear viscoelasticity. This simulated dataset is created via the experimental setup in Sections 5.1 and 3 , and the constitutive parameters are drawn from the prior. Note that the resolution of the image data visualized here is much lower than that of the actual data used in the numerical study; see an image snapshot with high resolution in Figure 17 . The material deformation is discernible yet small due to the narrow range of the imposed strain ( <5%<5\% ) suitable for linear viscoelasticity.
Figure 5: The surrogate FIM testing accuracy as a function of training sample size for designing uniaxial tests of linear and nonlinear viscoelastic materials in Sections 5 and 6 , respectively. The solid and dashed lines indicate the mean relative errors for predicting the log. FIM (measured in the Frobenius norm) and the approximate information gain, respectively. The surrogate prediction of the approximate information gain is obtained from the surrogate FIM predictions using 16 .
Figure 6: The optimized designs of uniaxial test for reliable learning of linear viscoelasticity found by EGA and ESFIM maximization. ( Left ) Comparison of the EGA values at random, EGA-optimized, and ESFIM-optimized designs. ( Right ) Visualization of the optimized designs found via EGA maximization and ESFIM maximization.
Figure 7: Visualizations and statistics of the 95% CIs for constitutive parameters of linear viscoelasticity inferred from experiments with random, ESFIM-optimized, and EGA-optimized designs. ( Top ) The box plots of the 95% CI sizes. The size distributions arise from the prior-based variation in the experimental outcomes. For random designs, the distributions also account for design variations. ( Bottom ) The relative expected reduction of the 95% CI sizes by optimized designs relative to random designs. The 95% CI sizes are estimated for θ\theta , which is transformed as in Section 5.1 from the physical parameters used as labels.
Figure 8: The normalized EGA design utility for learning marginal parameters of linear viscoelasticity grouped by material behaviors. ( Left ) The bar plots of the utility of random, ESFIM-optimized, and EGA-optimized designs for learning elasticity, anisotropy, viscosity, and relaxation. The error bars for the random designs represent the standard deviations of EGA with respect to the design variables. In contrast, the error bars for the optimized designs represent the standard errors of EGA attributed to finite parameter sample sizes. ( Right ) The expected improvement in the utility of optimized design relative to random designs for learning different behaviors.
Figure 9: Relation between principal-stress-based stress-state entropy and EGA for randomly selected designs and the EGA-optimized design. The EGA-optimized design also has the largest stress-state entropy, with an overall rank correlation of 0.670.67 .
Figure 10: The optimal batched designs of uniaxial testing for learning linear viscoelasticity found by ESFIM maximization. The designs in the top panel vary in geometry within their batches, whereas those in the bottom panel share the same geometry.
Figure 11: Visualization and statistics of the EGA design utility values of batched designs for uniaxial testing of linear viscoelastic materials. ( Left ) The EGA values for random and ESFIM-optimized (batched) designs of 1–3 experiments. ( Right ) Relative expected improvements of the EGA for the optimized designs with batch sizes of 1–3 compared to random designs with batch sizes of 1–3.
 Physical ParameterDescriptionBound
Equilibriumlog⁡μ∞\log\mu_{\infty}Log. of eq. shear modulus(-1.5, -0.5)
ν\nuInitial Poisson ratio(0.45, 0.49)
ϕf\phi_{f}Eq. fiber stiffness-to-shear log. ratio(0, 3)
log⁡k2\log k_{2}Log. of eq. fiber stiffening rate(-1, 1)
αf\alpha_{f}Fiber orientation angle(- π\pi /4, π\pi /4)
Non-Equilibriumqμ,iq_{\mu,i} , i=1,2i=1,2Viscous-to-eq. shear log. ratio(-1.5, 0.7)
Table 2: The bounds for the uniform distributions of the physical parameters in the re-parameterized nonlinear viscoelastic constitutive model.
Figure 12: An example of simulated force and image data for learning nonlinear viscoelasticity. This simulated dataset is generated using the experimental setup in Sections 5.1 and 3 , with an order-of-magnitude higher imposed strain. The constitutive parameters are drawn from the prior. Note that the resolution of the image data visualized here is much lower than that of the actual data used in the numerical study; see an image snapshot with high resolution in Figure 17 .
Figure 13: Visualization and analysis of ESFIM-optimized designs for uniaxial testing of nonlinear viscoelastic materials. ( Top ) The designs obtained via ESFIM maximization with batch sizes of 1–3. ( Bottom Left ) The EGA design utility values for random and ESFIM-optimized designs with batch sizes of 1–3. ( Bottom Right ) The expected improvements in the EGA of the optimized designs with batch sizes of 1–3 relative to random designs with batch sizes of 1–3.
Figure 14: The experimental designs with batch sizes 1–3 found via ESFIM maximization for uniaxial testing of nonlinear viscoelastic materials. The designs within the same batch share a geometry, and the design optimization is subject to strain-rate regularization in 24 with three regularization constants λ∈{0.25,1,4}\lambda\in\{0.25,1,4\} .
Figure 15: Visualizations and statistics of the 95%95\% CIs for constitutive parameters of nonlinear viscoelasticity inferred from experiments with random and ESFIM-optimized designs. We consider design optimization without ( λ=0\lambda=0 ) and with ( λ=0.25\lambda=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 designs, the distributions also account for design variations. ( Bottom ) The expected reduction of the 95% CI sizes by optimized designs relative to random designs. The sizes are estimated for θ\theta , which is transformed as in Section 6.1 from the physical parameters used as labels.
Figure 16: The normalized EGA design utility for learning marginal parameters of nonlinear viscoelasticity grouped by material behaviors. We consider design optimization without ( λ=0\lambda=0 ) and with ( λ=0.25\lambda=0.25 ) strain-rate regularization. ( Left ) The bar plots of the utility values of random and ESFIM-optimized designs for learning matrix equilibrium, fiber, viscous amplitude, and viscous kinetics properties. The error bars for the random designs represent the standard deviations of the utility with respect to the design variables. In contrast, the error bars for the optimized designs represent the standard errors of EGA due to finite parameter sample sizes. ( Right ) The expected improvement in the utility of optimized design relative to random designs for learning different properties.
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 17: High resolution snapshots of image data. ( Left ) The last snapshot in Figure 4 for uniaxial testing of linear viscoelastic materials. ( Right ) The last snapshot in Figure 12 for uniaxial testing of nonlinear viscoelastic materials.
Figure 18: Randomly sampled designs used for NMCE evaluations.
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