Shape-morphing soft materials can enable diverse target morphologies through voxel-level material distribution design, offering significant potential for various applications. Despite progress in basic shape-morphing design with simple geometries, achieving advanced applications such as conformal implant deployment or aerodynamic morphing requires accurate and diverse morphing designs on complex geometries, which remains challenging. Here, we present a Spectral and Spatial Neural Operator (S2NO), which enables high-fidelity morphing prediction on complex geometries. S2NO effectively captures global and local morphing behaviours on irregular computational domains by integrating Laplacian eigenfunction encoding and spatial convolutions. Combining S2NO with evolutionary algorithms enables voxel-level optimisation of material distributions for shape morphing programming on various complex geometries, including irregular-boundary shapes, porous structures, and thin-walled structures. Furthermore, the neural operator's discretisation-invariant property enables super-resolution material distribution design, further expanding the diversity and complexity of morphing design. These advancements significantly improve the efficiency and capability of programming complex shape morphing.

翻译：形变软材料可通过体素级材料分布设计实现多样化的目标形态，在各领域具有重要应用潜力。尽管在简单几何体的基础形变设计方面已取得进展，但实现诸如共形植入物部署或气动形变等高级应用，仍需在复杂几何体上进行精确且多样化的形变设计，这仍具挑战性。本文提出一种谱空间神经算子（S2NO），能够在复杂几何体上实现高保真形变预测。S2NO通过集成拉普拉斯本征函数编码与空间卷积，有效捕捉不规则计算域上的全局与局部形变行为。将S2NO与进化算法相结合，可在多种复杂几何体（包括不规则边界形状、多孔结构和薄壁结构）上实现形变编程的体素级材料分布优化。此外，神经算子的离散不变性特性支持超分辨率材料分布设计，进一步拓展了形变设计的多样性与复杂性。这些进展显著提升了复杂形变编程的效率与能力。