Computerized structural optimization methods are often used to design the shape of structures to achieve a desired function, such as a specific compliance. In the simplest case of a structure built from a material with small deformations obeying Cauchy elasticity, the compliance is constant and the relationship between the forces applied on the structure and its deformation is linear. This linearity severely limits the types of functions which can be achieved by such materials. Here we study mechanical metamaterials made of repeating unit cells, each with specific dimensional parameters and with one-sided contact non-linearities. We show that the force and displacement equilibrium configurations of these metamaterials are mathematically equivalent to the fixed points of a recurrent artificial neural network. By exploiting this equivalence, we demonstrate mechanical metamaterials that can be designed (trained) to implement complex non-linear functions, using a gradient descent algorithm as in artificial neural networks. One of our metamaterial structures has a higher compliance when it is pressed against a pattern of raised bumps corresponding to the vowels in the Braille alphabet, than when it is pressed against patterns for six consonants. As artificial neural networks are known to be efficient models for numerous problems in machine learning, our results reveal that beneficial features of neural networks can be transferred to physical objects (mechanical structures). These features include the design of systems with complex input–output relationships by using generic methods that only rely on the repetitive processing of pairs of inputs and desired outputs, as well as remarkable generalization capabilities. We anticipate our methodology to be a starting point for the transfer of some of the breakthroughs in the rapidly advancing field of machine learning to highly functional physical devices in applications that are constrained by energy, volume, data processing or response time.

计算机结构优化方法通常用于设计结构形状以实现所需功能，例如特定的顺应性。在由服从柯西弹性的小变形材料建造的结构的最简单情况下，柔量是恒定的，并且施加在结构上的力与其变形之间的关系是线性的。这种线性严重限制了这种材料可以实现的功能类型。在这里，我们研究了由重复晶胞制成的机械超材料，每个晶胞都具有特定的尺寸参数和单侧接触非线性。我们表明，这些超材料的力和位移平衡配置在数学上等同于循环人工神经网络的不动点。通过利用这种等价性，我们展示了可以设计（训练）以实现复杂非线性函数的机械超材料，使用梯度下降算法，如人工神经网络。当我们的一个超材料结构被压在与盲文字母表中的元音相对应的凸起的图案上时，它比当它被压在六个辅音的图案上时具有更高的顺应性。众所周知，人工神经网络是机器学习中众多问题的有效模型，我们的结果表明，神经网络的有益特征可以转移到物理对象（机械结构）上。这些特征包括通过使用仅依赖于输入和期望输出对的重复处理的通用方法来设计具有复杂输入-输出关系的系统，以及卓越的泛化能力。我们预计我们的方法将成为将快速发展的机器学习领域的一些突破转移到受能源、容量、数据处理或响应时间限制的应用程序中的高性能物理设备的起点。