Bayesian optimal design accelerates discovery of material properties from bubble dynamics

TL;DR

The paper tackles efficient discovery of soft material viscoelastic properties under high strain-rate cavitation by coupling inertial microcavitation rheometry (IMR) with a Bayesian optimal experimental design (OED) framework that maximizes the expected information gain $EIG$. It integrates a forward bubble-dynamics model (dimensionless Keller–Miksis) with constitutive laws (NHKV, qKV, Gen. qKV) and uses an ensemble-based four-dimensional variational method (En4D-Var) for data assimilation and parameter inference, while Bayesian model selection discriminates among constitutive laws via marginal likelihoods. In simulations with synthetic PA hydrogels, the sequential design rapidly identifies the correct model and yields sub-percent level parameter errors within roughly 10 sequential experiments, with high confidence in model discrimination (e.g., $p(\mathcal{M}|\mathbf{Y}^D, \mathbf{d})$ favoring the true law). The approach offers a robust, data-efficient pathway to design cavitation experiments and infer high-strain-rate material properties, with potential applicability to tissue phantoms and laser-based manipulation where uncertainty and resource limits are critical.

Abstract

An optimal sequential experimental design approach is developed to computationally characterize soft material properties at the high strain rates associated with bubble cavitation. The approach involves optimal design and model inference. The optimal design strategy maximizes the expected information gain in a Bayesian statistical setting to design experiments that provide the most informative cavitation data about unknown soft material properties. We infer constitutive models by characterizing the associated viscoelastic properties from measurements via a hybrid ensemble-based 4D-Var method (En4D-Var). The inertial microcavitation-based high strain-rate rheometry (IMR) method ([1]) simulates the bubble dynamics under laser-induced cavitation. We use experimental measurements to create synthetic data representing the viscoelastic behavior of stiff and soft polyacrylamide hydrogels under realistic uncertainties. The synthetic data are seeded with larger errors than state-of-the-art measurements yet match known material properties, reaching 1% relative error within 10 sequential designs (experiments). We discern between two seemingly equally plausible constitutive models, Neo-Hookean Kelvin--Voigt and quadratic Kelvin--Voigt, with a probability of correctness larger than 99% in the same number of experiments. This strategy discovers soft material properties, including discriminating between constitutive models and discerning their parameters, using only a few experiments.

Figures (9)

• Figure 1: Schematic of the IMR-based sequential BOED. Given a modeling parameter, $\vb*{\theta}=\{\mathcal{M},\, \vb*{\phi}_{\mathcal{M}}\}$, which includes a constitutive model and its material properties, and a design $\vb*{d}$ that describes the experimental setup (for example, the equilibrium radius), the IMR approach numerically solved the spherically symmetric motion of bubble dynamics. In computation, the complete flow states $\vb*{q}$ include bubble radius, bubble-wall velocity, temperature, and other variables, but they are only partially observable and are denoted as $\vb*{y}$.
• Figure 2: BO output trajectories (a) and the optimal BO outputs (b) for EIG sample sizes $N_\mathrm{EIG}= 200, 400, \dots, 1600$. The dot-dashed line in (a) indicates the onset of the BO process with 10 initial trials. The relative difference between the optimal design, $\vb*{d}^*$, using $N_\mathrm{EIG}$ samples and the design using 1600 samples, $\vb*{d}_{\infty}^\star$, is shown in (b). The optimal EIG for the EIG sample size used later, $N_\mathrm{EIG} = 1000$, is highlighted in (b).
• Figure 3: DA outputs over the total En4D-var iteration number $N_\text{DA}$: ensembles for (a) $G_{\infty}$; (b) $\mu$; (c) $\alpha$; and (d) RMSE of the bubble dynamics curves (see examples in \ref{['f:model selection']}). The shaded area in (a) represents the $95\%$ confidence interval for the $G_{\infty}$ measurements. The solid curves in (a--c) represent Gaussian distributions approximated from the 48 ensembles, with their respective mean values marked as stars. In (d), the error between the mean of the measurements and the unobtainable truth, $\|\langle\vb*{Y}^{\mathrm{D}}\rangle-\vb*{\tilde{Y}}\|$, is shown for comparison.
• Figure 4: Posterior bubble dynamics trajectories and their marginal likelihoods: (a, c) $R_\text{max}=9.85d-4m$ and $R_{\infty}^*=0.2887$; (b, d) $R_\text{max}=3.87d-4m$ and $R_{\infty}^*=0.15$.
• Figure 5: Sequential BOED outputs over the design number $N_\mathrm{Des}$: (a) EIG and total EIG; (b) model probabilities; and (c) relative error of the mean material properties.