🔬 condensed matter

Metamaterials that learn to change shape

The paper introduces a new class of metamaterials that utilize a contrastive learning scheme to dynamically update their local stiffnesses, enabling them to learn, forget, and sequentially master complex, non-reciprocal, and multistable shape-changing behaviors for adaptive physical tasks like gripping and locomotion.

Original authors: Yao Du, Ryan van Mastrigt, Jonas Veenstra, Corentin Coulais

Published 2026-09-11

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CC BY 4.0

Original authors: Yao Du, Ryan van Mastrigt, Jonas Veenstra, Corentin Coulais

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Living things are masters of adaptation. A cell changes its form to divide; a tissue reshapes itself to heal; an animal alters its posture to navigate a new terrain. This ability to learn and change shape in response to the environment is a fundamental strategy of life. For decades, scientists have tried to build synthetic materials that mimic this biological flexibility. They have created "metamaterials"—engineered structures designed to bend, twist, and morph in specific ways. These materials have shown promise in fields ranging from medicine to architecture. Yet, a crucial piece of the puzzle has been missing. While these human-made structures can be programmed to take on a shape, they cannot learn. Once built, their behavior is fixed. They cannot look at a new challenge, adjust their internal properties, and figure out a new way to move. They lack the capacity to adapt after they are made.

Researchers at the University of Amsterdam and its partners have now bridged this gap by creating metamaterials that can learn. Instead of being hard-wired with a single function, these new materials can be shown examples of a desired movement and then adjust their own internal stiffness to achieve it. The team built a robotic chain made of small units connected by an elastic skeleton. Each unit contains a motor and a tiny computer that can measure how much it has bent and talk to its neighbors. By applying a specific learning rule, the material can be shown a target shape, such as a curve or a letter, and then gradually tweak the stiffness of its joints until it can form that shape on its own when triggered. This process allows the material to forget old shapes and learn new ones, or even learn to perform complex, non-reversible movements that would be impossible for a standard material.

The core of this discovery lies in a method called contrastive learning. Imagine holding a flexible chain in your hands. To teach it a shape, you first let it hang freely while you push one end to a specific angle; this is the "free state." Then, you hold both the pushed end and the desired final shape in place; this is the "clamped state." The material's internal computers compare the difference between how the chain naturally bent in the first step and how it was forced to bend in the second. Based on this difference, the material automatically adjusts the stiffness of its joints. It makes the joints stiffer or softer in just the right places so that the next time it is pushed, it naturally settles into the desired shape. This happens without a central brain telling every part what to do; each unit only needs to know about its own movement and the movement of its immediate neighbor.

In their experiments, the researchers demonstrated this with a chain of six units. They showed the material how to form the letter "U" by applying a specific push. After just ten rounds of this comparison and adjustment process, the material learned the shape perfectly. When they pushed the input again, the chain curled into a "U" without any external help. The team then tested the limits of this learning ability. They used a longer chain of eleven units and taught it to spell out the word "LEARN," letter by letter. Remarkably, the material could forget the previous letter and learn the next one without needing to be reset or reprogrammed from scratch. It simply updated its internal memory of stiffness to accommodate the new shape.

The researchers also explored whether this learning could handle more complex tasks, such as learning multiple shapes at once or performing movements that break the usual rules of symmetry. In the physical world, forces often work both ways: if you push point A to move point B, pushing point B usually moves point A in a predictable, mirrored way. The team found that by introducing a special type of "active" stiffness that could act differently depending on the direction of the push, their material could break this symmetry. They trained a chain to curl one way when pushed from the left, but curl the opposite way when pushed from the right. This non-reciprocal behavior allowed the material to learn multiple, conflicting shapes simultaneously. For instance, the same chain could be trained to form the letter "L" when one specific unit was pushed, and the letter "E" when a different unit was pushed, all at the same time.

Perhaps the most surprising finding was that these materials could learn to have more than one stable shape. Usually, if you bend a spring and let go, it snaps back to its original flat position. However, by adjusting the learning rules to allow for negative stiffness in certain joints, the team created a material that could settle into two different stable shapes. Once pushed, it would jump to a new configuration and stay there, even after the push was removed. This bistability gave the material robotic capabilities. They built a small gripper that could automatically catch a moving object and hold it, then release it with a kick. They also created a chain that could walk. By training the chain to have four stable shapes and applying a simple, rhythmic push to a single joint, the entire chain began to crawl across a table. The non-reciprocal nature of the learning created a loop in the material's movement, allowing it to move forward in a continuous cycle using only one motor.

These findings suggest that metamaterials can serve as a powerful platform for physical learning, moving beyond static designs to adaptive systems. The researchers showed that this learning works even with simple rules and does not require high-precision sensors or complex processors. While the current experiments were conducted on flat, two-dimensional chains, the principles are general enough to be extended to three-dimensional structures. The ability to forget, relearn, and perform complex, non-symmetrical movements opens the door to a new generation of adaptive materials and soft robots that can evolve their behavior in real-time, much like the living organisms that inspired them.

Technical Summary: Metamaterials that Learn to Change Shape

Problem Statement
Living organisms possess a fundamental capacity to adapt and evolve their shape-changing responses through local modifications of their components. While synthetic metamaterials have successfully demonstrated advanced shape-morphing capabilities across various scales (from microns to meters), they typically lack the ability to learn. Traditional shape-memory materials can be trained to memorize only one or two static shapes, and most metamaterials are designed "once and for all" without the capacity to adapt their mechanical response to changing conditions or to learn new behaviors after fabrication. The central challenge addressed in this work is the creation of synthetic materials capable of physical learning—specifically, the ability to learn, forget, and relearn complex shape-changing responses in response to mechanical inputs.

Methodology
The authors construct a robotic metamaterial composed of $N$ units connected by an elastic skeleton. Each unit is equipped with a motorized hinge capable of exerting torque and a microcontroller. The system operates on a low-friction air table, allowing free movement.

Control Mechanism: The local stiffness of each unit is tunable via a local feedback loop. The torque $\tau_i$ applied to unit $i$ is governed by:
$\tau_i = -(k^o_i + k_e)\delta\theta_i - (k^p_{i-1} + k^a_{i-1})\delta\theta_{i-1} - (k^p_i - k^a_i)\delta\theta_{i+1}$
where $k^o_i$ is on-site stiffness, $k^p_i$ is passive (symmetric) neighbor stiffness, $k^a_i$ is active (anti-symmetric) neighbor stiffness, and $k_e$ is the fixed skeleton stiffness. The variables $\delta\theta_i$ represent angular deflections.
Contrastive Learning Scheme: The system employs a supervised physical learning protocol based on contrastive learning. The learning degrees of freedom are the tunable stiffnesses ( $k^o, k^p, k^a$ ), while the physical degrees of freedom are the angular deflections. The learning process involves two states:
1. Free State: Only input deformations ( $\delta\theta_I$ ) are imposed.
2. Clamped State: Both input and desired output deformations ( $\delta\theta_O$ ) are imposed.
The system updates stiffness parameters using a local learning rule derived from the gradient of the difference between a function $\psi$ evaluated in the clamped (C) and free (F) states:
$\frac{dk_i}{dt} = -\gamma \frac{\partial}{\partial k_i} (\psi^C - \psi^F)$
- Reciprocal Case ( $k^a_i = 0$ ): $\psi$ is the elastic energy. The learning rule updates $k^o$ and $k^p$ based on the difference in squared angles and cross-terms between states.
- Non-Reciprocal Case ( $k^a_i \neq 0$ ): To handle path-dependent systems where forces do not derive from a scalar potential, the authors define a path-dependent work function $\psi$ that includes an anti-symmetric term weighted by a path-direction factor $\alpha_i$ . This allows the learning of non-reciprocal responses where the force transmission is not symmetric.
Multistability: To achieve multistable configurations, a local stability constraint based on the Gershgorin circle theorem is introduced. This modifies the learning rule to include a term driving the on-site stiffness $k^o_i$ toward a predetermined value $k^*$ , allowing the system to learn unstable (bistable) configurations where the flat state is no longer a stable equilibrium.

Key Results

Sequential Learning and Forgetting: A metamaterial with $N=6$ units successfully learned to morph into a "U" shape from a straight chain in response to a specific input. Crucially, the system demonstrated the ability to "forget" this shape and sequentially learn new shapes (forming the word "LEARN" with $N=11$ ) without reinitialization.
Scalability: Numerical simulations up to $N=10^3$ indicated that while learning becomes harder with system size due to elastic decay, scalability can be maintained by introducing longer-range interactions or multiple outputs. Experimental validation with a 48-unit system successfully morphed into a cat shape using three inputs.
Non-Reciprocal Learning: The authors demonstrated that reciprocal metamaterials ( $k^a_i=0$ ) fail to learn shape changes that break symmetry between loading directions. By enabling non-reciprocal interactions ( $k^a_i \neq 0$ ) via the path-dependent learning rule, the metamaterial successfully learned non-reciprocal responses (e.g., positive curvature at unit 2 causing positive curvature at unit 5, but positive curvature at unit 5 causing negative curvature at unit 2).
Multi-Target Learning: Non-reciprocity enabled the simultaneous learning of multiple incompatible shape changes. While reciprocal systems failed to learn more than one target, non-reciprocal systems successfully learned up to three targets (and four with next-nearest-neighbor interactions), allowing a single material to form different letters ("LEREN") based on different input locations and angles.
Multistability and Robotic Functionality: By training for negative stiffness values, the metamaterials learned bistable configurations. This capability was utilized to create:
- Reflex Gripper: A device that automatically catches a moving object upon contact and releases/kicks it away upon a specific input, leveraging the bistable nature of the units.
- Locomotion: A chain trained to generate the letter "M" with four stable shapes. When driven by a single unit with a sine torque, the combination of multistability and non-reciprocity induced a cyclic shape shift, resulting in unidirectional locomotion on a substrate.

Significance and Claims
The paper establishes metamaterials as a viable platform for physical learning. The authors claim that by systematically adjusting internal parameters (learning degrees of freedom) using local learning rules, synthetic materials can emulate the adaptive shape-changing capabilities of living organisms.

The significance of this work lies in:

Overcoming Design Limitations: Moving beyond "design-once" materials to systems that can learn, forget, and relearn complex responses on demand.
Physical Learning Generalization: Demonstrating that physical learning schemes can generalize to energy non-conserving (non-reciprocal) and nonlinear (multistable) cases, which are essential for complex robotic functionalities.
Emergent Robotics: Showing that these learned properties enable autonomous robotic behaviors such as reflex gripping and locomotion without complex external control algorithms, relying instead on the material's intrinsic learned mechanics.

The authors conclude that this approach paves the way for adaptive materials and distributed robotics, with future avenues including the extension to dynamical/stochastic scenarios and unsupervised learning to further emulate autonomous living matter.

Technical Summary: Metamaterials that Learn to Change Shape

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