Metamaterials that learn to change shape

TL;DR

The paper presents metamaterials that learn to change shape through a local physical learning rule, advancing beyond fixed-design shape morphing. It introduces a path-dependent contrastive learning framework that uses a contrast between free and clamped mechanical equilibria to update learning degrees of freedom via $\\frac{d k_i}{d t}=-\\gamma\\frac{\\partial}{\\partial k_i}(\\psi^{C}-\\psi^{F})$, with a path-sensitive work function $\\Delta W$ and a generalized $\\psi$ that includes antisymmetric, nonreciprocal interactions. By enabling nonreciprocity through $k_i^{a}$ (and extended next-nearest couplings), the system learns multiple targets, forms nonreciprocal shape changes, and exhibits multistability that supports robotic reflex gripping and locomotion. Stability analyses (linear and nonlinear) and Gershgorin-based constraints provide a framework to control monostable versus multistable behavior, while simplified binary variants show that hardware-constrained implementations remain feasible. Collectively, the results establish physical learning in metamaterials as a scalable path toward adaptive, shape-programmable materials and soft robotics with nonreciprocal and multistable capabilities.

Abstract

Learning to change shape is a fundamental strategy of adaptation and evolution of living organisms, from bacteria and cells to tissues and animals. Human-made materials can also exhibit advanced shape morphing capabilities, but lack the ability to learn. Here, we build metamaterials that can learn complex shape-changing responses using a contrastive learning scheme. By being shown examples of the target shape changes, our metamaterials are able to learn those shape changes by progressively updating internal learning degrees of freedom -- the local stiffnesses. Unlike traditional materials that are designed once and for all, our metamaterials have the ability to forget and learn new shape changes in sequence, to learn multiple shape changes that break reciprocity, and to learn multistable shape changes, which in turn allows them to perform reflex gripping actions and locomotion. Our findings establish metamaterials as an exciting platform for physical learning, which in turn opens avenues for the use of physical learning to design adaptive materials and robots.

Figures (16)

• Figure 1: Contrastive learning for shape-changing metamaterials.a, Contrastive learning scheme. In the free state, the system is deformed from its initial equilibrium state by the input angle $\delta\theta^{I}$, whereas in the clamped state, both the input $\delta\theta^{I}$ and the desired output $\delta\theta^{O}$ are kept fixed. During learning, steps (ii-iv) are repeated while the learning degrees of freedom are updated according to the contrastive learning rule until a predetermined number of epochs is reached. b, The mean squared error (MSE) curves in simulation (solid line) and experiment (red dots) where a $N=6$ robotic chain is trained to morph into a U-shape. Here, the learning rate is $\gamma=0.01$. See the simulation protocol in the Methodology. c, Equilibrium configurations of each epoch in the free state. Note that the two edge units are not actuated. d, The stiffness matrix $K$ during learning. $K_{ij}$ refers the entry on $i^{\text{th}}$ row and the $j^{\text{th}}$ column in the stiffness matrix. Note that $K$ is a tridiagonal matrix since only the nearest-neighbor interaction is involved here. The initial parameters are $k_{i}^{o}=0.1$, $k_{i}^{p}=0.01$ and $k_{i}^{a}=0$. Note $k^{e}$ is a constant and thus not shown. e, A metamaterial with $N=11$ is sequentially trained to form the word "LEARN". See Extended Data Fig. \ref{['figE:continue learning']} for the corresponding MSE curves. The red linkage applies the input angular deflection.
• Figure 1: The side view of the robotic unit cells. Each unit cell is a motorized vertex connected by 3D printed plastic arms and elastic rubber bands. It consists of a DC motor embedded in a cylindrical heatsink and a microcontroller connected to a custom electronic board. The electronic board enables communication between vertices. Each motorized vertex sits on top of a red circular disk that ensures that the robotic unit floats on the air table. We apply external deformations by manually fastening the screws.
• Figure 2: Learning non-reciprocal shape changes and multiple targets.a, The procedure of learning non-reciprocal shape changes. Each target shape is learned following the above protocol in Fig. \ref{['fig:Learning demenstration']}a but the learning is conducted by switching between these two targets in turn during each epoch. b, The MSE curves of learning the above non-reciprocal shape changes in the $p$ configuration (red, Eq. \ref{['eq:tau_i']} with $k_{i}^{a}=0$) and the $a$ configuration (blue, Eq. \ref{['eq:tau_i']} with $k_{i}^{a}\neq0$) show that these targets can only be learned simultaneously with non-reciprocal interactions, i.e., in the $a$ configuration. Due to human operation error and the precision limitation of the experimental setup, the experimental MSE deviates slightly from the simulated curve after 10 epochs. c, The stiffness matrix $K$ of the metamaterial in the $a$ configuration during learning. The initial parameters are $k_{i}^{o}=0.1$, $k_{i}^{p}=0.01$ and $k_{i}^{a}=0$. The learning rate is $\gamma=0.05$. d, Simulation results of learning multiple targets with (non)reciprocal, and next nearest neighbor interactions, i.e., the $pp$ (Eq. \ref{['eqM:tau_aa']} with $k_{i}^{a}=k_{i}^{aa}=0$) and $aa$ (Eq. \ref{['eqM:tau_aa']} with $k_{i}^{a},k_{i}^{aa}\neq0$) configurations. A system of $N=10$ is simulated and the number of targets $N_{T}$ is varied from 1 to 8. The black semi-transparent dots are the MSE of each simulation and each column consists of 500 simulations. The solid line is the average MSE. The cut-off of the MSE is arbitrarily chosen to be $10^{-5}\ \mathrm{rad}^2$.
• Figure 2: The MSE curves of learning to form the word “LEARN” sequentially in Fig. \ref{['fig:Learning demenstration']}e. It shows that metamaterial can forget the previous shape change and relearn the next one without requiring reinitialization. Here, the learning is conducted in simulation and the learning rate is $\gamma=0.01$. The initial parameters are $k_{i}^{o}=0.1$, $k_{i}^{p}=0.01$ and $k_{i}^{a}=0$.
• Figure 3: Learning multistable shape changes and robotic functionalities. a, The normalized work landscape of unit 3 (yellow dot) by tuning $\delta\theta_3$. Two local minima correspond to two stable configurations, the letters "W" and "N". b, A pair of units shows bistable behavior. The flat configuration corresponds to zero deformations. Upon perturbation, the system jumps to a non-zero deformation instead of springing back to the initial configuration. The left inset shows the two stable configurations. The right inset shows the force fields of the bistable case in which two stable fixed points exist. The colorbar shows the normalized total torque $||\mathbf{F}||/ ||\mathbf{F}||_{\mathrm{max}}$ where $\mathbf{F}=\{\tau_{1}, \tau_{2}\}^{\top}$ and $\tau_{i}$ is the torque of unit $i$. c, The real part of the eigenvalues $\lambda$ during learning for a pair of units in both monostable and bistable scenarios that learn the same target. The desired shape change is generating $\delta\theta_{2}=-\pi/6$ rad after applying $\delta\theta_{1}=\pi/6$ rad. The imaginary part is zero so it is not shown here. d, A metamaterial with $N=6$ is trained as a reflex gripper (see Video S4). It can automatically catch a moving object and release it when an input is applied. e-g, Using a trained metamaterial with non-reciprocal interactions to achieve locomotion (see Video S4). e, The metamaterial with $N=5$ initially learns to generate the letter "M" (shape 1) and has four stable shapes. The system is driven by applying an external sine torque $\tau_{4}^{\mathrm{ext}}$ on unit 4 (yellow dot). f, The deformation of the system over time with the $p$, $a$ configuration and when it locomotes with the $a$ configuration. Data plotted in shape space projected onto the two basis vectors ($P_{1}$, $P_{2}$) (Eq. \ref{['eqM:cyclic_vectors']}) and colored with time. g, Snapshots of the locomotion and the trajectory of the center of mass colored by the angle of the projected shape $P_1+iP_2$. With the airtable inclined under an angle, gravity $\vec{g}_{\text{eff}}$ points downwards.