🔬 applied physics

In situ local learning of dynamic network materials

This paper introduces an in situ local learning framework for dynamic network materials that utilizes forward and time-reversed adjoint drives to enable the physical system to autonomously compute gradients and train its own mechanical parameters, thereby allowing the material to acquire diverse dynamical functions directly from tasks without pre-designed structures or expert knowledge.

Original authors: Shuaifeng Li, Xiaoming Mao

Published 2026-09-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Shuaifeng Li, Xiaoming Mao

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

Matter has long been viewed as a passive stage upon which forces act, or as a static tool built to perform a single, fixed job. A bridge holds weight; a spring stores energy; a lens focuses light. In each case, the function is hard-coded into the material's shape and composition during manufacturing. If the environment changes or a new task arises, the material cannot adapt. This limitation has driven a growing interest in a different kind of matter: one that learns. Just as a biological brain adjusts its connections to recognize a face or a sound, researchers are exploring whether the physical bonds and masses inside a material could be tuned to perform complex tasks, from hiding objects from waves to sorting sounds, simply by experiencing the right kind of feedback. This approach treats the material not as a finished product, but as a system capable of acquiring intelligence through its own motion.

A team of researchers at the University of Michigan has now demonstrated a way to teach these dynamic materials to learn new functions by applying a specific two-step physical protocol. They developed a method called "in situ local learning," which allows a physical network of springs and masses to calculate its own mistakes and adjust its internal properties directly within the physical system. Instead of relying on a digital twin to tell the material how to change, the researchers use the material itself to compute the necessary adjustments. The process involves two distinct phases of motion. First, the material is driven by an input signal, such as a vibration or a force, and its response is measured. If the response is not what was desired, an error signal is generated. This error is then recorded, reversed in time, and fed back into the same physical network as a second, backward-driving signal. By observing how the material reacts to this time-reversed error, the system can identify exactly which internal connections need to be strengthened or weakened to improve its performance.

The beauty of this method lies in its simplicity and locality. The researchers do not need to measure the entire state of the network or perform complex calculations in a central computer. Instead, each individual spring and mass only needs to know about the forces and movements happening right next to it. When the forward signal and the time-reversed error signal meet, they interact locally to produce a gradient, a mathematical direction that tells each part of the network whether to become stiffer, heavier, or more resistant to motion. This allows the material to update its own stiffness, mass, and internal friction simultaneously, effectively "learning" a new behavior through its own physical dynamics. The researchers tested this framework on a disordered network of mechanical nodes and bonds, a structure that might look like a tangled web of springs and weights, and showed that it could master four very different tasks simply by changing the definition of the desired outcome.

In the first demonstration, the material learned to act as a cloak for waves. A stiff, heavy obstacle was placed in the middle of the network, a feature that would normally scatter incoming waves and create a chaotic mess downstream. The researchers set a goal for the material: make the waves passing the obstacle look exactly as if the obstacle were not there at all. By repeatedly applying the forward and backward signals, the network adjusted the stiffness and mass of the springs surrounding the obstacle. The result was a broadband cloak that worked across a range of frequencies, suppressing the scattering signature and allowing waves to flow past the obstruction as if it were invisible. This was achieved not by following a pre-written blueprint, but by the material discovering the necessary configuration through trial and error, guided only by the error signal.

The second task required the material to perform a feat that is usually impossible for ordinary lenses: seeing the fine details of an object that are smaller than the wavelength of the wave used to see it. In standard physics, these tiny details are carried by "evanescent" waves, which fade away almost immediately and never reach a distant detector. The researchers trained the network to act as a lens that could capture this fading information and transport it to an image plane. The material learned to manipulate the near-field structure of the waves, preserving the high-frequency details that usually vanish. When tested with different letter shapes, the trained network successfully reconstructed the images, proving that it had learned to harness the invisible, decaying parts of the wave field to create a clear picture.

The third example showed that the material could learn to use friction, or damping, to its advantage. In most mechanical systems, damping is seen as a nuisance that drains energy and stops motion. Here, the researchers asked the network to do the opposite: to use a fixed amount of damping to create a large, sharp burst of motion at a specific moment. By redistributing the friction across the network, the material learned to organize its internal vibrations so that they interfered constructively, creating a powerful transient spike at the output. This enhancement was not possible with uniform or standard damping, proving that the material had learned a sophisticated, non-standard way to manage energy dissipation to achieve a specific goal.

Finally, the researchers applied the same learning rule to a task that resembles machine learning: classifying sounds. They fed the network audio waveforms of three different vowel sounds and asked it to sort them into three categories. The network was not programmed with the rules of phonetics or signal processing. Instead, it was simply told which output region should light up for each sound. Through the same local learning process, the material adjusted its internal parameters to recognize the unique temporal patterns of each vowel. It successfully learned to direct the energy of the correct sound to the correct output channel, achieving high accuracy in distinguishing the vowels. This demonstrated that the same physical system could function as a classifier, processing time-dependent data and making decisions based on its learned internal state.

These results suggest a new paradigm for how we design and use materials. Rather than engineering a specific structure for a specific job, we can provide a material with a learning rule and a goal, and let it find the solution on its own. The framework bridges the gap between physical materials and artificial intelligence, showing that the laws of physics can be used to perform computation and learning directly in the hardware. The material does not just respond to the world; it learns from it. While the current demonstrations rely on the principles of physical backpropagation that can be implemented in real hardware, the approach offers a route to materials that can adapt to changing environments, repair their own functions, or acquire new skills after they have been built. The ability to learn temporal responses in situ means that the next generation of smart materials might not just be programmed by engineers, but might learn to program themselves.

Technical Summary: In Situ Local Learning of Dynamic Network Materials

Problem Statement
Functional materials traditionally rely on pre-designed structures to realize specific functionalities, whereas physical neural networks (PNNs) transform physical inputs into outputs through native dynamics. A significant gap exists between these paradigms: current approaches often require "in silico" training of a differentiable digital twin followed by transfer to hardware, which suffers from a simulation–reality gap due to fabrication disorder, drift, and modeling uncertainties. Furthermore, existing physical learning schemes are often limited to static equilibrium states, steady-state harmonic responses, or restricted regimes that omit inertia and damping. There is a lack of a general, local learning rule for dynamic network materials that operate in the full time domain, where responses are shaped jointly by stiffness, mass, and dissipation. The challenge is to enable materials to acquire functions directly from dynamical tasks through their own physical dynamics without relying on a global digital model.

Methodology
The authors introduce an in situ local learning framework for dynamic network materials governed by second-order differential equations. The core mechanism involves applying two distinct drives to the same physical network:

Forward Drive: An input signal $F(t)$ drives the network, generating a forward trajectory $u(t)$ , bond elongations $e(t)$ , velocities $\dot{u}(t)$ , and accelerations $\ddot{u}(t)$ .
Time-Reversed Adjoint Drive: An error signal, derived from the mismatch between the actual output and a target trajectory, is time-reversed and applied as a source to the same network. This generates an adjoint field $u_{adj}(t)$ .

The learning rule is derived from the adjoint-state principle. By defining a time-integrated loss function $L = \int_0^T \ell[u(t), t] dt$ , the gradient of the loss with respect to local parameters (spring constants $k$ , nodal masses $m$ , bond damping $b$ , and nodal damping $\beta$ ) is computed via local products of the forward and adjoint fields. Crucially, the adjoint equation for time-domain dynamics is mathematically equivalent to the forward equation with a time-reversed source, allowing the physical system to compute its own gradients.

The update rules are strictly local:

Bond parameters ( $k, b$ ): Require only co-located forward elongation $e_j$ , elongation rate $\dot{e}_j$ , and adjoint elongation $e_{adj,j}$ .
Nodal parameters ( $m, \beta$ ): Require only co-located forward velocity $\dot{u}_i$ , acceleration $\ddot{u}_i$ , and adjoint displacement $u_{adj,i}$ .

This approach eliminates the need for a differentiable surrogate of the entire device, shifting the burden from model fidelity to the design of physical protocols that expose local correlations.

Key Results
The framework was validated through four distinct demonstrations, each utilizing the same learning rule but different loss functions:

Broadband Wave Cloaking: The network was trained to suppress the scattering signature of a stiff, heavy obstacle. By minimizing the mean-squared error between the downstream response of the cloaked system and an obstacle-free reference, the network learned heterogeneous distributions of stiffness, mass, and damping. The trained material successfully acted as a broadband cloak over a frequency band, generalizing to unseen frequency-phase combinations without a closed-form transformation design.
Evanescent-Wave Imaging: The network was trained to map near-field object patterns (letters C, L, T, U, O, X) to an image plane, preserving high-spatial-frequency information that typically decays. The learned parameters created a lens that utilized evanescent components to achieve high target correlation and focal accuracy. Performance degraded when the object-lens gap increased or when high-spatial-frequency content was removed, confirming the mechanism relies on near-field transfer.
Passive Transient Output Enhancement: With fixed stiffness and mass, the network redistributed a fixed passive damping budget to enhance a transient displacement at a specific output node following a force pulse. The trained non-proportional damping distribution significantly outperformed uniform damping and proportional (Rayleigh) damping controls. Analysis revealed that the enhancement arises from non-normal modal dynamics, where non-orthogonal eigenvectors allow decaying modes to interfere constructively.
Vowel Classification: The network functioned as a physical neural network for temporal classification. Time-dependent acoustic waveforms of three vowels (/iy/, /ae/, /ei/) were encoded as mechanical inputs. The network learned to concentrate accumulated mechanical energy in specific output regions corresponding to the vowel class, achieving ~86% accuracy on a held-out test set.

Significance and Claims
The paper claims to provide a unified route for dynamic network materials to act as both programmable matter and physical neural networks. The primary significance lies in demonstrating that a single, general in situ learning rule can program diverse physical functions—from wave control and imaging to signal processing and classification—simply by changing the loss function, rather than relying on expert knowledge or task-specific design rules.

Key contributions include:

Extension of Physical Backpropagation: Moving from static or steady-state harmonic regimes to full time-domain dynamics, incorporating inertia and dissipation as trainable degrees of freedom.
Local Physical Learning: Establishing that gradients for time-domain tasks can be computed locally using only co-located forward and adjoint fields, making the approach compatible with the principle of physical learning where global state measurement is impractical.
Dissipation as a Resource: Demonstrating that damping is not merely a parasitic loss but a trainable resource that can be spatially redistributed to shape transient responses and modal interactions (e.g., creating non-normal dynamics for transient enhancement).
Reprogrammability: Showing that the same physical material can be retrained for different tasks (e.g., from focusing to absorption) via continued local learning.

The authors maintain a modest stance regarding experimental realization, acknowledging that current demonstrations assume access to dynamical fields (displacements, velocities, accelerations) that must be measured or inferred in hardware. They note that future implementations must address hardware noise, drift, and the physical adjustability of parameters, but the framework provides a theoretical foundation for materials that learn temporal responses in situ.

Technical Summary: In Situ Local Learning of Dynamic Network Materials

More like this