Bayesian-guided inverse design of hyperelastic microstructures: Application to stochastic metamaterials

Abstract

From a given pool of all feasible design variants, our aim is to identify a structure that achieves a target macroscopic stress response. For each candidate design, the response is obtained from a high-fidelity oracle, in particular, time- and resource-intensive computational homogenization or experiments. We consider the case where (i) the geometry cannot be conveniently parameterized, rendering gradient-based optimization inapplicable, and (ii) brute-force evaluation of all candidates is infeasible due to the cost of oracle queries. To tackle this challenge, we propose a Bayesian-guided inverse design framework that proceeds as follows. First, the dimensionality of the design variants is reduced through statistical feature engineering, and the resulting low-dimensional descriptors are mapped to effective constitutive parameters describing the macroscopic hyperelastic response. This mapping is modeled using a multi-output Gaussian process surrogate that accounts for correlations between the parameters. The surrogate is trained using uncertainty-driven active learning under severe budget constraints, allowing only a very limited number of high-fidelity oracle evaluations. Based on surrogate predictions, a finite number of promising candidates are shortlisted. Since the surrogate accuracy is inherently limited, the final selection of the optimal design is performed through high-fidelity oracle evaluations within the shortlist. In numerical test cases, we consider a dataset of 50,000 candidate structures. Active learning requires labeling less than half a percent of the full dataset. Bayesian-guided inverse design under unseen loading conditions reaches a prescribed error threshold with only a handful of oracle evaluations in the majority of cases.

Figures (10)

• Figure 1: Schematic overview of the proposed Bayesian-guided discrete inverse design framework. A data-efficient surrogate model guides the exploration of a large candidate design space toward a shortlist of promising structures, while high-fidelity physics-based evaluations ultimately decide the final design.
• Figure 2: Flowchart of the methodological framework: feature engineering produces reduced microstructure descriptors, uncertainty-driven active learning trains the GP surrogate with minimal labeled data, and Bayesian-guided inverse design uses the trained model to identify a feasible design while minimizing oracle evaluations.
• Figure 3: Representative subset of 25 randomly selected stochastic metamaterial unit cells from a dataset of 50,000 samples, highlighting the diversity of the generated designs.
• Figure 4: Sampling of the five loading paths in invariant space. The panels show pairwise relationships between $(I_1-3)$, $(I_4-1)^2$, and $(I_6-1)^2$ for the sampled deformation states obtained from the incompressible plane-stress deformation gradient.
• Figure 5: Active learning curve showing the MAE over the hold-out test set as a function of the number of observed microstructures. The error decreases rapidly and saturates after approximately 200 labeled designs.