Generative models struggle with kirigami metamaterials

Abstract

Generative machine learning models have shown notable success in identifying architectures for metamaterials - materials whose behavior is determined primarily by their internal organization - that match specific target properties. By examining kirigami metamaterials, in which dependencies between cuts yield complex design restrictions, we demonstrate that this perceived success in the employment of generative models for metamaterials might be akin to survivorship bias. We assess the performance of the four most popular generative models - the Variational Autoencoder (VAE), the Generative Adversarial Network (GAN), the Wasserstein GAN (WGAN), and the Denoising Diffusion Probabilistic Model (DDPM) - in generating kirigami structures. Prohibiting cut intersections can prevent the identification of an appropriate similarity measure for kirigami metamaterials, significantly impacting the effectiveness of VAE and WGAN, which rely on the Euclidean distance - a metric shown to be unsuitable for considered geometries. This imposes significant limitations on employing modern generative models for the creation of diverse metamaterials.

Figures (8)

• Figure 1: Mechanical metamaterials based on straight cuts. a Initial $6\times6$ pattern with alternating cuts. This architecture gives rise to auxetic behavior through the rotating squares mechanism. b Perturbation of the initial architecture is performed by adding rotations $\beta_{i,j}$ to each cut. The absolute value of rotations is capped by parameter $\beta_{max}$. c Resulting admissible design without intersections between cuts. The likelihood of obtaining intersection-free sample through random rotations ($\beta_{max}=90^\circ$) does not exceed 0.001%
• Figure 2: Suitability of the Euclidean Distance for two cuts. a Three different configurations (A: [$5^\circ$,$4^\circ$], B: [$-5^\circ$,$-3^\circ$] C: [$65^\circ$,$-45^\circ$]) of adjacent cuts with unit length between centers and length of $\sqrt{3}$. b The design space for the considered system with two cuts. Dark blue zones correspond to the angle pairs of intersecting cuts. Magenta and green lines show two possible routes between A and B. c Sequence of cut positions corresponding to direct transition from A and B (magenta path). d Sequence of cut positions for detour path shown by green line. Note passing configuration C on a route from A to B.
• Figure 3: Generative approaches. a Variational Autoencoder (VAE), comprised of Encoder and Decoder stages, learns to map the designs into latent space and retrieve them back. b Generative Adversarial Network (GAN) utilizes competition between Generator and Discriminator to create samples that look real. c Denoising Diffusion Probabilistic Model (DDPM) employs sequential addition of noise to map the designs to latent space.
• Figure 4: Limiting the perturbations enables control over the success rate of generation. a The average number of intersections in the samples and the likelihood of generating unit cells without intersections vs maximum deviation $\beta_{max}$ from the base structure. b Exemplary unit cells for a maximum added rotations $\beta_{max}$ of $20^\circ$, $60^\circ$ and $90^\circ$.
• Figure 5: Training of models for $\beta_{max}=20^\circ$. a The evolution of the average number of intersections during training for unit cells generated by different machine learning approaches. An averaging over five epochs was used for curve smoothing. b Distribution of the cuts with the specific added angles $\beta_{i,j}$ in the training dataset for $\beta_{max}=20^\circ$ (red) and in the set generated by trained DDPM (blue).