Generative Inverse Design of Metamaterials with Functional Responses by Interpretable Learning

Abstract

Metamaterials with functional responses can exhibit varying properties under different conditions (e.g., wave-based responses or deformation-induced property variation). This work addresses the rapid inverse design of such metamaterials to meet target qualitative functional behaviors, a challenge due to its intractability and non-unique solutions. Unlike data-intensive and non-interpretable deep-learning-based methods, we propose the Random-forest-based Interpretable Generative Inverse Design (RIGID), a single-shot inverse design method for fast generation of metamaterial designs with on-demand functional behaviors. RIGID leverages the interpretability of a random forest-based "design$\rightarrow$response" forward model, eliminating the need for a more complex "response$\rightarrow$design" inverse model. Based on the likelihood of target satisfaction derived from the trained random forest, one can sample a desired number of design solutions using Markov chain Monte Carlo methods. We validate RIGID on acoustic and optical metamaterial design problems, each with fewer than 250 training samples. Compared to the genetic algorithm-based design generation approach, RIGID generates satisfactory solutions that cover a broader range of the design space, allowing for better consideration of additional figures of merit beyond target satisfaction. This work offers a new perspective on solving on-demand inverse design problems, showcasing the potential for incorporating interpretable machine learning into generative design under small data constraints.

Figures (18)

• Figure 1: Schematic diagram of the RIGID method. We first train a random forest on a design-response dataset to learn the forward design-response relation — predicting qualitative responses (e.g., bandgap existence at any given wave frequency) of designs. Then given a design target, we can infer the likelihood of any design satisfying the target by probing into the trained random forest. New designs with tailored responses can be generated by sampling the design space based on the likelihood estimation.
• Figure 2: The inverse design pipeline of the proposed RIGID method (using the inverse design of acoustic metamaterials as an example). Given design parameters $\mathbf{x}$ and the auxiliary variable $s$ (e.g., wave frequency), a trained random forest predicts the probability of the qualitative response $y$ (e.g., bandgap existence). Each tree in the random forest splits the joint space of $\mathbf{x}$ and $s$ into regions, each associated with a specific prediction (shown on leaf nodes). The splitting criteria are encoded in tree nodes. "T" means meeting a criterion and "F" means not meeting it. RIGID first identifies leaf nodes that are relevant to the considered range of auxiliary variable $s$ by checking splitting criteria related to $s$ and pruning tree branches that are irrelevant (Step 1). If the considered range of $s$ has multiple parts, we repeat this step for each part and take the intersection of relevant leaves (Step 2). Each relevant leaf node corresponds to a decision path indicating a region in the design space, as well as a predicted probability of target satisfaction, which is a score we assign to the corresponding design space region (Step 3). With multiple trees in a random forest, we can average the scores predicted by each tree and use the average score as our likelihood estimation (Step 4). We can then sample from the design space based on the likelihood distribution to generate new designs tailored to the target (Step 5). Note that the 2-dimensional likelihood maps are only for visualization purposes. The actual dimension will be the same as the design space dimension (i.e., the number of design variables).
• Figure 3: Acoustic metamaterial design problem configuration and results. (A) Design variables of center and corner mass radii ($r_\text{center}$ and $r_\text{corner}$) and strut radius ($r_\text{strut}$). (B) High symmetry points of the cubic irreducible Brillouin zone. (C) A sample dispersion relation and bandgap (marked by the highlighted zone). The design objective is to generate new acoustic metamaterial designs with target bandgaps. (D) KDE of the estimated likelihood for generated designs. (E) Satisfaction rates, average scores, and selection rates for RIGID designs under varying sampling thresholds (solid lines), in comparison to the satisfaction rate of GA designs (horizontal dashed line). The horizontal dotted line indicates 100% satisfaction. (F) Geometries and corresponding dispersion relations of five RIGID designs with the highest likelihood of satisfying a specified target bandgap (marked as highlighted frequency regions). All the radii ($r_\text{strut}$, $r_\text{center}$, and $r_\text{corner}$) have a unit of $\mu$m. Here only the fourth design fails to meet a small portion (at around 6 MHz) of the target bandgap, whereas the others meet it. Generated designs for other targets are shown in SI Appendix, Figs. S2-S6.
• Figure 4: Distributions of satisfactory solutions for two bandgap targets. The off-diagonal plots show the pairwise bivariate distributions of design variables, and the diagonal plots show the marginal distributions of the data in each column. The left panel shows that GA designs are highly localized while RIGID can lead to diverse solutions. The right panel indicates that none of the GA designs satisfy the target, while satisfactory RIGID designs are diverse and can be very different from feasible designs from data. Solutions from data include feasible designs in both training and test data.
• Figure 5: Optical metasurface design problem configuration and results. (A-B) Design variables (materials, layer thicknesses, and cross-section geometry types). (C) A sample absorbance spectrum and the wavelength intervals (highlighted wavelength regions) corresponding to absorbance above the threshold $t$. The design objective is to generate new optical metasurface designs that exhibit higher absorbance than a threshold $t$ at the user-defined wavelength interval(s). (D) KDE of the estimated likelihood for generated designs. (E) Satisfaction rates, average scores, and selection rates for RIGID designs under varying sampling thresholds (solid lines), in comparison to the satisfaction rate of GA designs (horizontal dashed line). (F) Designs (geometries and material selections) and corresponding absorbance spectra of five metasurfaces generated by RIGID. These five solutions are generated designs with the highest likelihood of satisfying specified target high-absorbance regions (marked as highlighted wavelength regions). All the layer thicknesses ($h_l, l=1, 2, 3$) have a unit of nm. Here all five designs satisfy the target. Generated designs for other targets are shown in SI Appendix, Figs. S7-S9. (G) Distributions of design variables for satisfactory solutions generated by RIGID and GA (for the same target defined in Panel F). GA designs are highly localized and lack diversity compared to RIGID designs.