In-memory phononic learning toward cognitive mechanical intelligence

Abstract

Mechanical intelligence seeks autonomous materials that embody memory, perception, learning, and decision-making within the mechanical domain. A fundamental challenge is to integrate memory with information processing in an efficient, scalable, and interpretable way that supports cognitive learning and decision-making. Here, we introduce in-memory phononic learning, a framework that unifies non-volatile mechanical memory with wave-based perception in a phononic metastructure. Spatial information is stored as structural states that program elastic-wave responses, allowing probing waves to act as queries that retrieve informative features through parallel, physically transparent localization modes. Learning is achieved by offline optimization of input waveforms to selectively combine these features for multitask memory-pattern recognition without hardware redesign, with decisions inferred directly from output-energy responses. We experimentally and numerically demonstrate high-accuracy classification across tasks from geometric shapes to handwritten digits. Overall, this work elevates “materials that compute” toward “cognitive matter” that can interpret, reason, and learn through wave physics, laying a foundation for scalable and robust autonomous material systems.

INTRODUCTION

Mechanical intelligence (MI) (1–5)—the concept of designing intelligence within the mechanical domain of materials and structures—has recently emerged as a direction to enable autonomous engineering systems with high performance, efficiency, security, and resilience. Instead of outsourcing intelligence entirely to electronics and computers, MI envisions autonomous materials that store, process, and perceive information, and adapt to inputs through their intrinsic mechanical responses, thereby reducing latency and energy overhead while improving robustness under extreme conditions and resistance to cyber threats. These unique features and advantages make MI particularly critical for the realization of emerging complex engineering systems demanding advanced autonomy, such as smart robotics, adaptive infrastructure, automated vehicles, and personal wearables.

The foundation of MI lies in mechanical information processing and learning (6–9), which requires a memory module to retain knowledge from the environments, a computing module to process, interpret, and perceive information, and a physical communication interface that bridges the stored memory and computation. A widely adopted strategy emulates the digital computing paradigm by abstracting mechanical bits from multistable structures for non-volatile memory storage and implementing logic operations via carefully designed quasi-static state transitions and snap-through interactions (10–19). However, scaling such logic-based architectures to higher-level computing tasks–especially cognitive learning and adaptation—requires the sequential cascading of numerous basic logic components via complex mechanical coupling, where information is transferred between modules through slow quasi-static signal propagation, limiting both operational efficiency and architectural scalability.

Learning capability naturally demands substantially greater computing power, requiring systems to store and reason about information, perceive features, and make decisions in uncertain environments. Inspired by artificial or biological neural architectures, morphological computing or physical neural networks including mechanical (20–24) or wave-based neural networks (25–29) and physical reservoir computing (PRC) (30–35) have emerged to pursue learning using architected metamaterials. These approaches support higher-level cognitive features, such as machine-learning regression and classification tasks, through distributed physical transformations while avoiding explicit logic cascades. Yet, these systems often depend either on sophisticated structural optimization for task-specific training or on complex nonlinear dynamic reservoirs requiring external electronic readouts. Critically, learned input-output mappings emerge from opaque black-box correlations within disordered and intractable structural responses, undermining reliability in scenarios demanding transparent reasoning and preventing the establishment of predictive and transferable design rules–factors crucial for robustly deploying MI in real-world applications. What is missing, therefore, is a unified, efficient, and physically transparent framework that seamlessly couples memory with other elements of intelligence within the mechanical domain, in which stored information natively drives computation and perception, enabling learning and decision-making through interpretable and predictable mechanisms.

Here, we bridge the abovementioned gap by introducing an in-memory phononic learning framework that integrates non-volatile mechanical memory with wave-based information processing. This approach harnesses elastic wave propagation in phononic metastructures as an efficient and mechanistically explicit physical communication interface that directly connects memory and information processing for cognitive learning tasks. Phononic metastructures (36, 37) are an ideal medium for this role. Through rich wave dispersion, interference, and resonance, they can perform complex analog computations at the speed of sound while supporting parallelism across frequencies and modes (38–40). In this work, these features are leveraged to interact with information stored in non-volatile mechanical memory, with elastic waves serving as wave-based queries that retrieve memory-defined phononic features for downstream learning and decision-making through wave–memory interactions.

To realize and evaluate this framework, we develop a multistable phononic metastructure testbed that encodes complex spatial information from environmental inputs–such as payload or deformation patterns–into reconfigurable, stable metastructure states as non-volatile mechanical memory (Fig. 1A). Wave-based mechanical perception and cognitive learning capabilities are then investigated through exemplar pattern-recognition tasks, where structured elastic wave propagation is harnessed to perform in-situ classification of these encoded memory patterns (Fig. 1B). Specifically, the key feature of this testbed, as detailed in the Results section, is that it functionally separates memory-state switching from wave propagation while coupling them through controllable, state-dependent interactions. It enables the testbed to concurrently achieve effective wave transmission and state switching via different materials in different layers. As a consequence, the metastructure memory directly and deterministically programs the system’s elastic wave-propagation landscape, transforming the stored information into rich vibrational modes as phononic features through the formation of elastic waveguides. In situ perception occurs in the frequency domain by harnessing vibrational localization modes: under different harmonic excitations, the memory-defined geometry localizes wave energy into distinct, frequency-selective local waveguides, for example, along specific edges or corners, which act as spatial phononic “filters” to extract key features of the stored information. In this sense, probing waves at different frequencies serve as queries to interrogate the stored mechanical memory, selectively extracting informative geometric cues encoded in the stored information that can be combined for subsequent learning and decision-making. We then establish a learning framework to classify stored memory patterns from the perceived features by applying input excitation waveforms in the form of linearly weighted harmonic signals (Fig. 1B). Instead of optimizing the structural geometry, the learning process optimizes the spectral weights of these input waveforms offline, allowing them to serve as trainable wave queries that selectively activate or suppress feature-modes associated with specific memory-pattern classes, thereby redistributing wave energy across the output channels. The classification decision is then inferred directly from the output energy distribution across channels, representing a physical analog of prediction probabilities. In principle, this architecture is inspired by PRC but differs in how the trainable linear combination is implemented. Rather than applying an external trainable digital classifier through a linear readout layer to the measured physical response, the trainable parameters are encoded in the probing-wave spectrum, and the resulting decision information is represented by the vibration-energy responses of the trained channels. The architecture can be interpreted as a wave-physics-informed analog neural network, in which individual frequency responses serve as fixed feature-extraction layers operating in parallel across frequencies (i.e., constituting a physical reservoir), while the superposition principle allows the input waveforms to act as a trainable linear output layer to selectively combine perceived features and perform classification (Fig. 1C). This framework physically integrates the information-processing chain, including memory encoding, perceptual feature extraction, and learned input-output mappings, through a single, fixed mechanical substrate. The process is efficient, inherently parallel, and physically interpretable, as each step is governed by predictable and traceable memory-wave interactions in a structured medium, avoiding the black-box nature of disordered morphological systems. Furthermore, the framework naturally exhibits versatile computing power for multitasking, where different tasks can be executed with the same structure by simply updating the input waveforms. Taken together, we refer to this framework as in-memory phononic learning to emphasize that information is stored in non-volatile structural memory states and processed in situ through their direct interactions with elastic waves, while the learnable parameters are encoded in the probing-wave spectrum and the learned mapping is physically manifested in the resulting phononic responses.

Fig. 1. In-memory phononic learning framework.

(A) Memory encoding in a multistable phononic metastructure. The two symmetric stable states of each module are abstracted as mechanical bits ‘0’ and ‘1’ and can be independently programmed to write spatial information as non-volatile mechanical memory. (B) In-memory phononic learning for pattern recognition. Trainable input waveforms (linearly weighted superpositions of harmonic tones) act as wave queries that probe class-specific, frequency-selective vibrational localization modes. The classification decision is inferred directly by comparing the output energies associated with the input channels, with the energy distribution across channels as a physical analog of class probabilities. (C) PRC-inspired interpretation of the phononic learning architecture. The metastructure’s frequency responses form a fixed, high-dimensional, and physically interpretable physical reservoir in which diverse localization modes perceive class-specific geometric features, while the trainable input waves act as a linear readout layer that combines the queried features for classification outputs.

We experimentally and numerically validate this in-memory phononic learning framework by achieving high-accuracy recognition across a series of 2D spatial pattern classification tasks. By unifying memory, perception, and learning through a physically interpretable and transparent mechanism, we elevate the ambition from “materials that compute” to the next level–achieving “cognitive materials” that learn and understand through their inherent wave physics and providing a foundation for scalable, deployable, and robust mechanically intelligent matter that can interpret, reason, and adapt to dynamic, unpredictable, and data-rich environments.

RESULTS

Metastructure design for programmable non-volatile mechanical memory and wave physics

As illustrated in Fig. 2A, a three-layer multistable phononic metastructure is constructed to functionally separate memory encoding from wave propagation while coupling them through controllable, state-dependent interactions. It comprises an inner phononic plate sandwiched between two outer multistable layers that provide non-volatile mechanical memory. The inner plate is a locally resonant phononic metamaterial consisting of periodic square units, each containing a central resonator mass connected to a square frame through four thin beams (Fig. 2B). This layer is purposefully fabricated from a low-damping material to preserve high-fidelity wave responses needed for wave-based information processing. The top and bottom layers comprise arrays of curved-beam bistable elements mounted in rigid frames and fabricated from soft materials to facilitate reliable state switching for mechanical memory encoding and storage. Each opposing bistable pair is aligned with one inner-layer resonator so that the three components form a single-cell multistable module (Fig. 2C), which functions as a mechanical memory bit that can be independently programmed to encode information. To physically link the stored memory to wave physics, permanent disk magnets are affixed to the center of each outer bistable element and to each inner-layer resonator mass. The magnets are axially magnetized and positioned to generate repulsive forces between the inner resonator and its two outer counterparts. Switching the states of the bistable pair changes the inter-magnet distance and thus modulates the local coupling. Here, we define the two symmetric stable states for each module as mechanical bits ‘0’ and ‘1’, where bit ‘0’ means both outer bistable elements are switched outward with an inter-magnet distance of $d_0=23 \text{mm}$, resulting in low magnetic coupling, and bit ‘1’ means both are switched inward with a reduced inter-magnet distance of $d_1=6 \text{mm}$, strengthening the magnetic coupling (see the dashed box in Fig. 1A). Figure 2E shows our experimental prototype of an 11-by-11 phononic metastructure (see Supplementary Materials, section S1 for details of the prototype fabrication and experimental setup). We experimentally validate the multistability of the testbed through quasi-static tensile testing on individual modules and demonstrate the programmable, non-volatile mechanical memory for information encoding at the array level by manually switching selected modules to write spatial patterns into the testbed (see Supplementary Materials, section S2 for details of the state-switching behavior of the metastructure modules, and examples of mechanical information encoding into the testbed).

Fig. 2. Multistable phononic metastructure design and modeling.

(A) Schematics of the three-layer phononic metastructure, containing a low-damping inner phononic metamaterial plate sandwiched between two multistable outer layers that provide non-volatile mechanical memory. (B) Inner phononic metamaterial layer consists of periodic square units with side length $a=3 \text{cm}$, each with a central resonator mass. (C) A multistable module with two outer curved-beam bistable elements mounted in a rigid frame and an inner locally resonant unit. Magnetized disk magnets are affixed to the outer elements and on the inner resonator to introduce memory-wave interaction. (D) Numerical model of the module. The inter-layer magnetic coupling is modeled as linear spring foundations, a translational spring $k_l$ and a rotational spring $k_{\mathrm{\theta }}$, connecting each inner resonator with fixed ground. (E) Front and side views of the 11-by-11-unit experimental prototype.

With the proposed platform, the state-dependent magnetic coupling reconfigures the local resonances and, consequently, the phononic properties of the inner metamaterial plate, providing the crucial link between stored memory and wave physics. This functional separation allows the static, non-volatile memory to remain encoded in the outer multistable layers, while the inner phononic layer provides a wave-based pathway for in situ interrogation and processing of the stored information without individually accessing or reprogramming the memory units.

To this end, we investigate the programmable wave physics of the metastructure under different memory states via finite-element simulations and experiments. Once a memory configuration is programmed, we consider small-amplitude out-of-plane elastic wave propagation about the corresponding equilibrium state. For numerical simulations, a model is established by linearizing the dynamics about each programmed equilibrium configuration and representing the magnetic coupling from the outer layers as equivalent linear spring foundations connecting each inner-layer resonator to fixed ground, which include a translational spring in the $z$-direction and a rotational spring about the $x$- and $y$-axes (Fig. 2D). The effective translational and torsional stiffnesses $k_l$ and $k_{\mathrm{\theta }}$ under different memory states are characterized through experimental measurements and numerical magnetic modeling. Under bit ‘1’, the coupling yields a positive translational stiffness of $2.5\times {10}^3 \mathrm{N}/\mathrm{m}$ and a negative torsional stiffness of $-0.011 \mathrm{N}·\mathrm{m}/\text{rad}$, whereas under bit ‘0’, both effective stiffnesses are negligible (see Supplementary Materials, section S3, for details of the magnetic coupling characterization and numerical model).

We first analyze the linear dispersion relation by considering a basic periodic unit to which the Floquet-Bloch periodic boundary conditions are applied. The Bloch wavevector $\mathbf{k}=(k_x,k_y)$ is swept along the edges of the 2D irreducible Brillouin zone (shaded area in Fig. 3D) within the reciprocal lattice. The unit-cell band structures under bits ‘0’ and ‘1’ are obtained, as plotted in Fig. 3A and B. In the considered frequency range from 0 to 600 Hz, the bit-‘0’ state exhibits two complete locally resonant bandgaps for the out-of-plane modes: one between 256 Hz and 339 Hz, and another above 566 Hz. For the bit-‘1’ state, the increased magnetic coupling dramatically alters the band diagram. The lowest out-of-plane branch, representing translational motion in the $z$-direction (Fig. 3C, star), shifts upward due to the positive translational stiffness, creating a new foundation-induced bandgap from 0 to 168 Hz. Concurrently, the other two out-of-plane branches, associated with rotational motion about the in-plane axes (Fig. 3C, diamond/triangle), shift downward due to the negative torsional stiffness. Experimental measurements validate this memory-dependent band structure tailoring, showing strong agreement with these results (see Supplementary Materials, section S4 for experimental validation of the bandgap programmability).

Fig. 3. Memory-programmed band structures.

Band structures for a periodic unit under bit ‘0’ (A) and bit ‘1’ states (B). Circles indicate in-plane modes (x-y-plane vibration), and filled dots represent out-of-plane modes (z-direction). Shaded regions highlight identified out-of-plane bandgaps. (C) Mode shapes (shown in displacement magnitude field $\mid \mathbf{d}\mid$) for the three out-of-plane branches. The lowest branch indicates translational modes (star), and the other two branches are rotational modes (triangle/diamond). (D) 2D irreducible Brillouin zone within the reciprocal lattice (shaded area). The wavevector $(k_x,k_y)$ is swept along the edges of the irreducible Brillouin zone ($\mathrm{\Gamma }\to X\to M\to \mathrm{\Gamma }$). (E) 2-by-2 and 3-by-3 supercell memory configurations. 30 independent configurations are identified, accounting for symmetry and periodicity equivalence. (F) Bandgaps across all configurations. The horizontal axis represents configuration indices, and the vertical bars show the out-of-plane bandgap frequency regions for each configuration. The bar color encodes the fraction of ‘1’ bits within each memory configuration.

We then generalize the analysis to periodic supercells by considering different spatial memory bit distributions within 2-by-2-cell and 3-by-3-cell units, which lead to more diverse wave behaviors. Accounting for periodicity and symmetry-induced equivalence, we identify 30 independent bit configurations (Fig. 3E). For each configuration, the corresponding bandgap frequency regions are calculated, as summarized in Fig. 3F, where the horizontal axis represents the indices of different memory configurations and the vertical bars show the corresponding bandgap regions. We observe rich memory-dependent wave characteristics where each configuration produces a unique band diagram. Even with an identical bit ‘0’/‘1’ ratio but different spatial distributions, the band diagrams can be entirely distinct and can generate additional band branches across different frequencies. Physically, different arrangements of mechanical bits modify local resonances and couplings between units, leading to configuration-dependent symmetry breaking and degeneracy lifting in diverse ways (see Supplementary Materials, section S5 for more details on 2-by-2 and 3-by-3 supercell dispersion analyses). This establishes a predictable and deterministic mapping between the encoded memory states and the resulting wave-propagation landscape. Fundamentally, it converts the memory into rich phononic information carried by the wave-transmission spectrum, which specifies the frequency channels conveying the stored information.

Frequency-selective wave energy localizations for memory-pattern feature perception

With the fundamental wave physics uncovered, we now investigate how these memory-programmed phononic properties enable a more sophisticated capability: frequency-selective wave energy localization that performs feature-level perception of stored information. While previous research has exploited rich waveguiding behavior in phononic metamaterials (36, 41–44), such as defect-induced energy confinement along specific paths or edge and corner modes induced by topological symmetry breaking, we uniquely utilize these phenomena to realize a mechanical perception system in which the wave energy localization modes selectively perceive and extract informative geometric features of the stored memory patterns.

Here, we exploit defect-induced wave energy localization, where the ‘defect’ arises from the mechanical bit reconfiguration during information writing. We first uncover the fundamental principle through dispersion analysis of a strip supercell containing a single programmed defect state. When a single unit within a uniform background is switched to a different memory bit, it creates a local defect with resonance perturbation that traps wave energy locally at specific frequencies. For instance, in a strip of uniform bits ‘0’, programming the central unit to bit ‘1’ (referred to as Type I encoding) introduces sharp defect modes within the original bandgap of the corresponding defect-free strip, as indicated by red dots in Fig. 4A, left, concentrating vibrational energy precisely at the defect unit. Conversely, in a strip of uniform bits ‘1’, programming the central unit to bit ‘0’ (referred to as Type II encoding) generates defect modes within the foundation-induced bandgap (Fig. 4A, right), with similar spatial localization but at entirely different frequencies. The existence of these frequency-selective defect modes is further validated through full-scale metastructure experiments and simulations, as shown in Fig. 4B, where different-frequency waves can be selectively guided and localized along defect paths by reconfiguring the metastructure memory states at excitation frequencies predicted by the eigenfrequency ranges of these defect-mode bands (see Supplementary Materials, section S6 for additional results on periodic strip supercell dispersion and experimental validation of defect modes). This frequency-selective localization provides a direct mechanism for processing the stored information: defect modes act as spatial phononic filters, producing locally enhanced responses at the programmed sites so that patterns encoded in the memory can be read from the scattering wavefields.

Fig. 4. Frequency-selective wave energy localization for feature perception.

(A) Band structures of a strip supercell with a central defect state under Type I and Type II encoding. The wavevector $k_y$ is swept from 0 to $\mathrm{\pi }/a$, where $a$ is the unit side length. Shaded regions highlight the bandgaps of the corresponding defect-free strip. A localization parameter $\mathrm{\Lambda }={\iiint }_{V_{\mathit{\text{defect}}}}{\mid \mathbf{d}\mid }^2\text{dV}/{\iiint }_{V_{\mathit{\text{total}}}}{\mid \mathbf{d}\mid }^2\text{dV}$ is defined to quantify the energy confinement at the defect unit, where $\mathbf{d}$ is the displacement field, and $V_{\mathit{\text{defect}}}$ and $V_{\mathit{\text{total}}}$ represent the volume of the central unit and the strip, respectively, with $\mathrm{\Lambda }\approx 1$ indicating defect modes (red bands). (B) Localization-mode-induced waveguiding in the metastructure through experiments and simulations. Two waveguides are programmed: a straight line of ‘1’ bits in a ‘0’ background and an L-shaped ‘0’ path in a ‘1’ background. At the defect-mode frequencies predicted in (A), steady-state out-of-plane velocity fields $v_z$ show energy localized and guided along the encoded paths. (C) Eigenfrequencies (left) and mode shapes (right) of the metastructure with letters ‘U’ and ‘M’ encoded as Type II defects. In the left plot, the x-axis represents mode index, and the y-axis indicates eigenfrequency. Mode shapes are shown as squared out-of-plane displacement magnitude $\mid d_z{\mid }^2$, normalized for visualizing energy distribution within the metastructure. Each encoded pattern generates diverse and frequency-selective defect modes that decompose the shapes into small-piece geometric characteristics, enabling feature-level perception. (D) Simulated frequency responses under harmonic excitations at the central unit. In the left plot, the y-axis represents excitation frequency, and the x-axis is the metastructure’s steady-state energy output. The right plots show scattering wavefields (squared out-of-plane velocity magnitude $\mid v_z{\mid }^2$) for the two patterns at 99 Hz and 126 Hz that selectively excite the corresponding defect modes and enable discrimination by probing wave responses. For each pattern, the scattering wavefields share the same color-bar scale.

This defect-mode physics scales elegantly to complex 2D patterns, enabling rich feature-level perception. As illustrated in Fig. 4C, when complex spatial information, such as letters ‘U’ and ‘M’, is encoded into the metastructure memory, the system generates a set of distinct and diverse defect modes that naturally decompose the patterns into constituent geometric features. Each defect mode localizes wave energy at specific locations, such as edges, sharp corners, or junctions, with the spatial wave energy distribution directly mapping to particular geometric characteristics of the encoded memory pattern. Crucially, these feature-modes resonate at distinct frequencies due to variations in local magnetic coupling and resonator interactions, which enables frequency-selective feature perception through spectral wave probing. Figure 4D presents the frequency response spectra under the two memory patterns by applying harmonic excitation at the central unit. It is observed that the ‘U’ pattern exhibits resonance peaks where its specific defect modes are activated, while the ‘M’ pattern displays a different set of peaks corresponding to its unique modes. Specifically, this spectral response separation enables pattern discrimination by probing the metastructure energy output at specific excitation frequencies. For example, as shown in Fig. 4D, right, a 99 Hz excitation activates one of the ‘U’ pattern feature modes, resulting in a clear energy-localized scattering wavefield with high energy output, while the ‘M’ pattern is off resonance with much lower energy output. Yet, the reverse occurs at 126 Hz (resonance for ‘M’), indicating the feasibility of harnessing the wave responses to uniquely identify and classify different stored information. We note that, in these two cases, the information is encoded as Type II defects with bits ‘0’ forming the patterns on a ‘1’ background. Supplementary Materials, section S7 provides examples of feature perception under Type I encoding.

As such, the phononic metastructure functions as a mechanical perception system in the frequency domain, where each vibrational localization mode acts as a specific feature detector, with the spatial distribution of wave energy indicating what geometric characteristic is being perceived and the corresponding resonance frequency providing the query channel through which this feature can be retrieved. It should be noted that, although wave propagation about each programmed configuration is governed by linearized dynamics, the mapping from the spatial memory pattern to the resulting phononic features is generally nontrivial and configuration-dependent. Each programmed pattern defines a distinct global dynamic operator, and the spatially distributed memory bits collectively reshape the resonance and localization landscape. This transformation of spatial memory patterns into identifiable phononic features through physically grounded wave localization establishes the foundation for the proposed in-memory phononic learning framework.

In-memory phononic learning for pattern recognition

Building upon the feature perception capabilities, we create an in-memory phononic learning framework that enables the metastructure to perform general pattern classification tasks, as illustrated in Fig. 1B. The core idea is to treat the fixed, memory-programmed metastructure as a physical feature extractor in the frequency domain, while optimizing only the spectral weights of the input waveforms, which serve as trainable wave queries that selectively activate and interrogate the most class-relevant feature modes. The classification decision is then obtained by comparing the resulting metastructure vibration-energy responses.

For an $N$-class classification task, we define $N$ input channels {$I_1,\dots ,I_N\}$. Each channel corresponds to one class and is a trainable linearly weighted combination of different harmonic tones in the form of $I_i\left(t\right)={\sum }_{j=1}^K{w}_{\mathit{ij}}\text{sin}\left(2\mathrm{\pi }{f}_{j}t\right)$, which forms a composite wave query for retrieving class-relevant phononic features from the encoded memory pattern. Here, {$f_1,\dots ,f_K\}$ are sampled excitation frequencies that cover the frequency regions where the corresponding defect modes occur, and $\left\{w_{\mathit{ij}}\right\}$ are trainable parameters that determine the contribution of each frequency component. For a given encoded memory pattern, we measure the metastructure’s steady-state response at each $f_j$ under the same excitation amplitude and compute an energy feature $E\left(f_j\right)$, i.e., the metastructure’s output energy under excitation frequency $f_j$ in terms of summed vibration energy over all resonators, which quantifies the feature information retrieved by that frequency-specific wave query. In the small-amplitude linear regime, the time-averaged output energy under channel $I_i$ can be calculated as $E_{\mathit{\text{out}}}^{\left(i\right)}\propto {\sum }_j{w}_{\mathit{ij}}^2 E\left(f_j\right)$, which is the nonnegative linear combination of energy features under a constant scaling factor across channels. This energy measure provides a consistent objective for training and a quantitative metric for performance evaluation.

We then normalize these energy outputs across all channels to obtain an $N$-dimensional probability vector $\mathbf{p}=[p_1,\dots ,p_N]$, where ${\mathbf{p}}_i=\frac{E_{\mathit{\text{out}}}^{\left(i\right)}}{{\sum }_{k=1}^N{E}_{\mathit{\text{out}}}^{\left(k\right)}}$. The class with the highest probability indicates the prediction. In practice, this probability vector is directly inferred by probing the energy output across the channels, where the highest-energy channel indicates the predicted class. The spectral weights $\left\{w_{\mathit{ij}}\right\}$ are optimized offline on labeled training samples using cross-entropy loss, where the energy features under harmonic excitations are precomputed as training data. ${\mathrm{\ell }}_1$ regularization is applied to promote sparse frequency selection and automatically identify the most discriminative feature modes for each class, so that each learned input channel selectively activates the localized vibrational modes associated with its target class while suppressing response for other classes. Full details of the framework formulation, dataset collection, training process, and implementation are given in Supplementary Materials, section S8.

In principle, while our framework is inspired by the architecture of PRC at a high level by implementing a wave-physics-informed analog neural network, as depicted in Fig. 1C, it is fundamentally different. Here, the frequency responses of the metastructure serve as the physical reservoir, mapping each encoded memory pattern into a rich, configuration-dependent, high-dimensional phononic feature space that inherently operates in parallel across frequencies. The trainable input waveforms then act as a linear output layer, combining these perceived features to perform classification. While standard PRC often relies on disordered systems to generate complex nonlinear dynamics needed for a rich feature space, our system uses rich and predictable wave physics from a structured metastructure medium. This crucial difference ensures that the high-dimensional feature mapping emerges from physically traceable and interpretable localization modes, rather than from the opaque, black-box dynamics. More specifically, the trainable linear combination is implemented through the spectral composition of the probing waves, rather than through an additional trainable digital classifier acting on the measured spatial response. The memory-dependent phononic dynamics then physically implement the learned mapping and generate class-dependent vibration-energy outputs, whose relative strengths determine the predicted class. This eliminates the need for an external electronically implemented linear readout layer commonly used in PRC, advancing toward truly self-contained cognitive mechanical intelligence.

Task 1: Geometric pattern classification

To validate the in-memory phononic learning framework, we first pursue a proof-of-concept task to classify geometric pattern shapes—squares, triangles, and crosses—encoded into the metastructure memory. These patterns are randomly generated with diverse scales and locations (Fig. 5, A and E). We evaluate both Type I (pattern as bits ‘1’) and Type II (pattern as bits ‘0’) encoding schemes using a dataset of 150 samples (50 per class), with 20% of the samples reserved for testing. The framework achieves 100% training and testing classification accuracy for both encoding types in simulation, as indicated in Fig. 5, C and G, where the output energy profiles from the well-trained input channels under each sample are plotted, revealing that the highest energy outputs always come from the targeted channel. To gain more physical insight, Fig. 5, B and F plot the frequency component weightings from the learned input channels for each case, as well as the corresponding mean frequency response among each class of patterns. It is noticed that, for each input channel, those most significant frequency components with higher weightings concentrate on well-isolated resonance peaks for the targeted class while being off resonance for the other two classes. It reveals a physics-grounded strategy of the proposed framework, where the training selects a sparse set of frequencies at which each input wave only activates the corresponding class-specific localization modes while suppressing the modes associated with the competing classes, and as a result, the output energy across channels is redistributed with optimally enhanced vibration energy level at the correct channel. This observation is further confirmed by examining the output wavefields of the metastructure under the well-trained input waves. For instance, Fig. 5, D and H visualize the scattering wavefields under the three well-trained channels for selected samples. We observe that $I_1$ activates square-related modes with energy localization at the diagonal corners of squares, while $I_2$ activates triangle-related modes along the triangle’s inclined edges, and $I_3$ selects cross-related modes concentrating energy at the intersection region. This demonstrates that the learning process automatically identifies and leverages the most geometrically discriminative features for classification. Experimental validation of the geometric pattern classification task is shown in Fig. 6, using a set of 54 training and 18 testing samples under Type I encoding. Figure 6A presents the measured steady-state wavefields for representative samples at six excitation frequencies that selectively activate each class-related feature mode, revealing the perception capability of the testbed. Figure 6B plots the normalized output energies across the trained input channels for all samples, where 53 out of 54 training samples and all testing samples are correctly recognized despite experimental noise, validating the effectiveness of the proposed framework. (See Supplementary Materials, section S9 for more details of numerical and experimental validations of geometric pattern classification tasks).

Fig. 5. Geometric pattern classification task.

Exemplar patterns for the three classes (square, triangle, cross) under Type I (A) and Type II (E) encoding. (B and F) Mean frequency response (left) for each class of patterns and the learned input waveform weightings (right). Dominant weights align with well-isolated class-specific resonance peaks and remain off resonance for untargeted classes, revealing a sparse, interpretable frequency-selection strategy. (C and G) Normalized output energy for all samples under the three input channels, with training and testing partitions indicated. Samples 1–50 correspond to squares, samples 51–100 are triangles, and samples 101–150 are crosses. Both encodings reach 100% simulation accuracy for both training and testing scenarios; Type I achieves lower classification confidence with 25 frequency components selected, while Type II achieves higher confidence with only 8 frequency components due to cleaner defect modes in the wide foundation-induced gap. (D and H) Scattering wavefields (steady-state time-averaged out-of-plane energy fields) for selected samples under the trained channels show energy localization at class-specific feature modes, e.g., diagonal-corner modes for squares under $I_1$, inclined-edge modes for triangles under $I_2$, and intersection modes for crosses under $I_3$. For each sample, the scattering wavefields share the same color-bar scale.

Fig. 6. Experimental validation of geometric pattern classification task.

(A) Measured steady-state responses (normalized squared out-of-plane velocity magnitude field, i.e., wave energy distribution within the metastructure) of representative samples under six excitation frequencies, showing frequency-selective wave energy localization that activates class-related feature modes. (B) Normalized output energy for all samples under the three input channels, with training and testing partitions indicated. Samples 1–24 correspond to squares, samples 25–48 are triangles, and samples 49–72 are crosses. 53 out of 54 training samples and all 18 testing samples are correctly classified.

Furthermore, by comparing the results from Type I and Type II encoding schemes, we find that the quality of wave localization directly impacts learning efficiency and confidence. Although both achieve 100% accuracy, we quantify the classification confidence by the average normalized energy in the targeted channels across all samples, which is equal to the sample-averaged predicted probability of the true class. Type II encoding, which produces strong, well-separated defect modes within a wide foundation-induced bandgap, requires only 8 frequency components across all input channels and leads to a high confidence of 0.98, indicating that nearly all vibration energy concentrates in the correct channels with minimal leakage to the untargeted ones. By contrast, for Type I encoding, the defect modes appear in the narrower locally resonant bandgap (see Supplementary Materials, sections S7 and S8 for more detailed analysis). This results in less distinct and noisier feature modes. Consequently, achieving the same accuracy requires a greater number of frequency components, and the output energy shows more diffusion into untargeted channels, yielding a lower confidence of 0.85. This underscores that a clearer physical mapping between memory and wave localization leads to more robust and efficient learning.

Task 2: Handwritten digit recognition task

To investigate generalizability, we apply the same metastructure to a more complicated and challenging task: classifying handwritten digits ‘0’ and ‘1’ from the benchmark MNIST dataset (45). We select 150 samples per digit, down-sampled from 28-by-28 grayscale images to 11-by-11 binary patterns for encoding into the metastructure (Fig. 7A), with 20% of the samples reserved for testing. Using Type II encoding, the system achieves a training accuracy of 98.3% and a testing accuracy of 93.3% in simulation by simply retraining the input waveforms (Fig. 7, B and C). Representative scattering wavefields under the trained input channels (Fig. 7D) reveal the underlying physics: $I_1$ activates loop-closing modes characteristic of digit ‘0’, localizing energy along a ring-like path (or a substantial arc), while $I_2$ excites edge-following modes aligned with the shape of a digit ‘1’. Despite substantial intra-class variation in thickness, curvature, and centering for handwritten digits, the classifier maintains strong performance in perceiving these class-specific feature modes, where 37 frequency components across channels are used in this task to facilitate shifts of defect-mode frequencies among different samples. The results demonstrate the multitasking capability of the framework, where the hardware remains unchanged and only the input excitations are updated–a key practical advantage over conventional morphological approaches that often require laborious, task-specific structural optimization and reconfiguration across different tasks. Furthermore, it demonstrates a more efficient pathway to scale computational complexity. Conventional morphological systems typically sequentially cascade multiple physical layers to enrich computing power, which increases architectural complexity and can compromise efficiency. In contrast, our framework scales within a single metastructure substrate by leveraging a richer set of frequency components. This essentially creates a high number of parallel, hidden layers in the frequency domain, where different frequency bands simultaneously extract diverse features. This inherent parallelism allows our system to handle more complex tasks like digit recognition without sacrificing its streamlined, single-layer architecture, while achieving high accuracy.

Fig. 7. Handwritten digit recognition task using the same metastructure.

(A) Example down-sampled digits ‘0’ and ‘1’ encoded as Type II patterns. (B) Training and testing confusion matrices. For each class, 120 samples are selected for training, and 30 samples are used for testing. 98.3% training accuracy and 93.3% testing accuracy are achieved. (C) Normalized output energy across samples under the two trained input channels. Samples 1–150 correspond to digit ‘0’, and samples 151–300 are digit ‘1’. The predicted label corresponds to the channel with maximal energy. (D) Scattering wave-energy fields under trained input waveforms for representative test digits. The ‘0’ channel ($I_1$) excites loop-closing localization around a ring-like path, while the ‘1’ channel ($I_2$) excites edge-following modes, yielding separability between digits ‘0’ and ‘1’. For each sample, the scattering wavefields share the same color-bar scale.

To further assess the information processing capacity of our platform, we extend the ‘0’/‘1’ handwritten digit classification task to a 14-by-14 metastructure in simulation. With increased spatial resolution and a larger dataset of 900 training and 100 testing samples per digit, the performance improves with reduced overfitting, achieving a testing accuracy of 96.0%. We then challenge the system with a full ten-class digit recognition task. For this highly complex task, we focus on highlighting the computing potential of the wave-based feature perception capability by employing a hybrid phononic-reservoir setup based on a PRC formulation, in which the metastructure generates the physical reservoir features and an external linear readout layer performs classification. For each harmonic excitation, the full spatial phononic wavefield is read out and concatenated into a feature vector. A linear readout layer is then trained to classify the resulting high-dimensional phononic feature representation. The purpose of this benchmark is to provide complementary evidence that the memory-dependent phononic responses form a rich and discriminative feature space, rather than to serve as a direct implementation of the proposed class-channel phononic learning framework. With 9000 training and 1000 testing samples, this setup achieves 96.1% testing accuracy, demonstrating that the phononic feature space perceived by the metastructure is sufficiently rich and discriminative for state-of-the-art tasks. Supplementary Materials, sections S10 and S11 provide additional numerical results and implementation details for the handwritten digit recognition tasks, including the 11-by-11 and 14-by-14 ‘0’/‘1’ digit classification and the ten-class hybrid phononic-reservoir benchmark.

The results from these exemplar tasks collectively show that our in-memory phononic learning framework successfully leverages the rich wave physics of structured media to achieve machine-learning classification tasks directly in the mechanical domain in an efficient and physically interpretable manner. More specifically, by programming wave-memory interactions in a single metastructure, we enable a form of mechanical cognition, where the metastructure can store encoded information and physically process and transform it into class-dependent vibration responses through learned probing-wave interactions. This establishes a physically embodied architecture for creating intelligent matter capable of adaptive reasoning and learning through its inherent physical properties.

DISCUSSION

In this research, we present an efficient, unified, and physically transparent framework for advancing the state of the art of mechanical intelligence–embodying in-memory phononic learning into architected materials that enable memory, perception, and learning capabilities, and we investigate its efficacy in and through a rationally designed phononic metastructure. By coupling non-volatile mechanical memory with wave-based information processing, we enable mechanical learning by harnessing memory-state-dependent wave dynamics as a physical analog neural network and, more specifically, as a wave-query interface to stored mechanical memory. It achieves effective feature-level perception through interpretable wave-localization modes and implements cognitive memory-pattern classification via trainable input waves that act as wave queries, integrating information encoding, perceptual feature extraction, and the physical implementation of the learned mapping through wave-memory interactions in a single metastructure. Utilizing example testbeds, our experimental and numerical efforts validate the framework’s effectiveness, where the same metastructure achieves high accuracy in classifying geometric shapes and handwritten digits, with multitasking capability enabled simply by updating the input waveforms. This approach overcomes key limitations in the state-of-the-art mechanical intelligence. First, it provides physical transparency–unlike black-box morphological systems, the learned mappings are directly traceable to physics-grounded phononic property tailoring. Second, it offers high efficiency through phononic parallel processing across frequencies within a single material layer, avoiding the need for complex cascading of basic logic modules or layers of physical substrates. Third, it possesses versatile computing power and generalizability to diverse tasks, eliminating the need for task-specific structural optimization or reconfiguration.

Ultimately, this research elevates the ambition from creating materials that merely compute to engineering cognitive matter that can sense, reason, learn, and understand through its intrinsic physical dynamics. These advances would support real-world applications including autonomous payload identification in adaptive infrastructures, tactile sorting in soft robotics, and embedded structural health monitoring where the material itself diagnoses damage patterns, opening new frontiers for the next generation of autonomous engineering systems with lower power consumption, more direct interaction with surroundings, better cybersecurity, and greater survivability in extreme environments. The core principles demonstrated here–using structured wave physics as a mechanism for querying and extracting information for in-memory processing and learning–are scalable and transferable across physical systems, such as photonic crystals for optical computing (46, 47), and acoustic devices for sonic communication and processing (48). Furthermore, this work offers an efficient, inherently parallel, and transparent hardware platform for the physical embodiment of machine-learning classification tasks, which could inspire new wave-based machine learning architectures with enhanced interpretability.

MATERIALS AND METHODS

Metastructure prototype fabrication and experimental setup

An 11-by-11 multistable phononic metastructure is fabricated via multi-material 3D printing and modular assembly. Quasi-static tensile testing is performed to validate module multistability and characterize inter-layer magnetic coupling. Frequency-response datasets are collected using an electromagnetic coil shaker for out-of-plane excitation and a scanning laser Doppler vibrometer for response measurements. Details of the geometry, materials, fabrication, and experimental setup are provided in Supplementary Materials, sections S1 to S4 and S6.

Simulation setup

Finite-element simulations are performed in COMSOL Multiphysics 6.2, including magnetic-field simulation and unit-cell, supercell, metastructure dispersion, and frequency-domain simulations. Material damping in the frequency-domain simulations is represented using an isotropic structural loss factor. Details of the numerical model, simulation setups, and damping sensitivity analysis can be found in Supplementary Materials, sections S3, S5 to S7, and S9.

Phononic learning framework

Training/testing datasets are precomputed from experiments or simulations. The probing-wave parameters are optimized offline in Python 3 using PyTorch 2.5.1. Detailed procedures can be found in Supplementary Materials, sections S8 to S11.

Supplementary Materials

This PDF file includes:

Supplementary Text Section S1 to S11

Figs. S1 to S16

Table S1

References