RAG: A Random-Forest-Based Generative Design Framework for Uncertainty-Aware Design of Metamaterials with Complex Functional Response Requirements

TL;DR

RAG presents a random-forest-based generative framework for uncertainty-aware inverse design of high-dimensional functional responses in metamaterials. By reformulating the forward problem to predict responses at arbitrary query points and quantifying uncertainty through an ensemble, it enables flexible, one-shot design generation conditioned on complex requirements without large datasets. The framework provides an explicit likelihood for design feasibility and uses Metropolis-Hastings sampling to draw diverse, feasible designs, with uncertainty guiding trust and risk. Demonstrations on acoustic dispersion and mechanical snap-through show data efficiency, diverse feasible solutions, and uncertainty-based filtering, highlighting potential for broader, small-data inverse-design applications beyond metamaterials.

Abstract

Metamaterials design for advanced functionality often entails the inverse design on nonlinear and condition-dependent responses (e.g., stress-strain relation and dispersion relation), which are described by continuous functions. Most existing design methods focus on vector-valued responses (e.g., Young's modulus and bandgap width), while the inverse design of functional responses remains challenging due to their high-dimensionality, the complexity of accommodating design requirements in inverse-design frameworks, and non-existence or non-uniqueness of feasible solutions. Although generative design approaches have shown promise, they are often data-hungry, handle design requirements heuristically, and may generate infeasible designs without uncertainty quantification. To address these challenges, we introduce a RAndom-forest-based Generative approach (RAG). By leveraging the small-data compatibility of random forests, RAG enables data-efficient predictions of high-dimensional functional responses. During the inverse design, the framework estimates the likelihood through the ensemble which quantifies the trustworthiness of generated designs while reflecting the relative difficulty across different requirements. The one-to-many mapping is addressed through single-shot design generation by sampling from the conditional likelihood. We demonstrate RAG on: 1) acoustic metamaterials with prescribed partial passbands/stopbands, and 2) mechanical metamaterials with targeted snap-through responses, using 500 and 1057 samples, respectively. Its data-efficiency is benchmarked against neural networks on a public mechanical metamaterial dataset with nonlinear stress-strain relations. Our framework provides a lightweight, trustworthy pathway to inverse design involving functional responses, expensive simulations, and complex design requirements, beyond metamaterials.

Figures (8)

• Figure 1: Illustration of inverse design on functional responses across different classes of metamaterials. In each case, the middle panel shows the functional response, while the bottom panel illustrates the corresponding design requirement. (a) Mechanical metamaterials: The functional response is the stress--strain relation, where stress $\sigma$ is a function of strain $\epsilon$. The design requirement is a prescribed nonlinear mechanical behavior, represented by the blue dashed line. (b) Acoustic metamaterials: The functional response is the dispersion relation, where frequency $\omega$ is a function of band index $i$ and wave vector $k$. The design requirements are indicated by blue and gray shaded areas, representing the desired passbands and stopbands, respectively. (c) Thermal metamaterials: The functional response is the temperature-dependent thermal properties, where thermal connectivity $\kappa$ is a function of temperature $T$. The design requirement---high conductivity at high temperatures and low conductivity at low temperatures---is illustrated by the blue dashed line. (d) Schematic illustrating the relationship among unit-cell geometry, functional responses, and design requirements in metamaterials design. The form of the design requirements could vary across applications and is often complex to specify.
• Figure 2: Formulation of forward mapping and inverse design, illustrating the relationship between (a) the unit cell geometry $\mathbf{x}$, (b) corresponding functional response $f$ and its discrete form $\mathbf{y}$, (c) design requirement $\mathcal{T}$, (d) desired functional response set $\mathcal{Y}^\mathcal{T}_\mathcal{G}$, and (e) design solution set $\mathcal{X}^\mathcal{T}$.
• Figure 3: Illustration of the RAG framework. (a) Step 1: forward prediction of $\mathbf{y}$ from $\mathbf{x}$ with uncertainty quantification (c) Step 2: uncertainty-informed inverse design given a design requirement $\mathcal{T}$.
• Figure 4: (a) Design space and response space in the 2d acoustic metamaterials in the case study. The unit cell structure is specified by three geometric parameters $\mathbf{x} = [r_\text{strut}, r_\text{center}, r_\text{corner}]$. (b) Forward prediction accuracy of random forests under different maximum tree depths, quantified by mean squared error (MSE). (c) Forward prediction with uncertainty for three testing samples. The uncertainty is quantified by the $\pm 2$ standard deviation denoted as $\pm 2 \hat{s}$. (d) Average uncertainty (quantified by $2 \bar{\hat{s}}$) of dispersion relations across different band orders.
• Figure 5: Inverse design results for acoustic metamaterials for two different requirements $\mathcal{T}_1$ (a) and $\mathcal{T}_2$ (b). The blue shaded area indicates the partial passband and the gray shaded area indicates the partial stopband. For each requirement, the top-5 highest likelihood unit cell geometries and the corresponding dispersion relations are displayed. For $\mathcal{T}_1$, all the five designs meet the requirement. For $\mathcal{T}_2$, none meets the requirement. The detailed specifications of all the ten requirements can be found in Table \ref{['tab: requirement list']}.