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.

翻译：面向先进功能的超材料设计通常涉及对非线性且条件依赖的响应（如应力-应变关系和色散关系）进行逆向设计，这些响应由连续函数描述。现有设计方法大多聚焦于向量值响应（如杨氏模量和带隙宽度），而功能响应的逆向设计因其高维性、在逆向设计框架中容纳设计需求的复杂性，以及可行解的不存在性或非唯一性，仍然具有挑战性。尽管生成式设计方法已显示出潜力，但它们通常需要大量数据、以启发式方式处理设计需求，并且可能在缺乏不确定性量化的情况下生成不可行的设计。为应对这些挑战，我们提出了一种基于随机森林的生成式方法（RAG）。通过利用随机森林的小数据兼容性，RAG能够以数据高效的方式预测高维功能响应。在逆向设计过程中，该框架通过集成估计似然度，从而量化生成设计的可信度，同时反映不同设计需求的相对难度。通过从条件似然中采样进行单次设计生成，解决了一对多映射问题。我们在以下案例中展示了RAG的应用：1）具有指定部分通带/阻带的声学超材料，以及2）具有目标突弹跳变响应的力学超材料，分别使用了500个和1057个样本。在一个包含非线性应力-应变关系的公开力学超材料数据集上，我们以神经网络为基准评估了其数据效率。我们的框架为涉及功能响应、昂贵仿真和复杂设计需求的逆向设计（不限于超材料）提供了一条轻量级且可信赖的途径。