Learning Associations in Reconfigurable Particle Packings via Local Cyclic Driving

Abstract

We investigate associative-memory behavior in a reconfigurable particle packing programmed by purely local cyclic driving. The system is a two-dimensional bidisperse Lennard--Jones particle assembly with periodic boundaries evolved under athermal quasistatic relaxation. During training, a fixed set of input particles is driven cyclically while output particles are selected on-the-fly by a region-driving rule and driven according to a prescribed flow pattern; during retrieval, only the inputs are driven. Associative-memory performance is quantified by the cosine similarity between realized and target output displacement directions. Unlike physical learning systems with fixed architecture, learning here arises through emergent weight updates: localized rearrangements modify the contact network and reshape the effective mechanical couplings between inputs and outputs. Across task difficulty we identify three regimes. In an easy setting, the intrinsic mechanical response already produces coherent motion in the right-hand region under input-only driving, yielding high performance without training. In a hard setting, the desired mapping conflicts with the dominant collective drift, resulting in low baseline performance and only modest training gains; introducing intermittent relaxation cycles reduces train--retrieval mismatch and improves performance. In an intermediate quadrupolar task, repositioning the input--output geometry stabilizes the desired response and converts initially stochastic trajectories into reproducible learned motions. Together these results identify minimal physical ingredients for association-based functionality in athermally driven particulate media and motivate an association learning phase diagram for reconfigurable matter.

Figures (10)

• Figure 1: Schematic of the training protocol, the retrieval protocol, and a performance metric. (a) The task is to associate two input particles (blue) with two output particles (green) to create a particular flow pattern in terms of direction. For this specific task, when the two blue input particles move closer together vertically, the green output particles both move downward. The training protocol is such that there are $N_r-1$ loading/unloading cycles in which both the inputs and the outputs are cyclically driven. In the unloading schematic, the lighter shading represent the initial positions of the particles. After $N_r-1$ loading/unloading cycles, for some tasks, a relaxation cycle is implemented in which the output particles are not driven. This process constitutes one training cycle. In addition, the filled black circles denotes that we choose two output coordinates initially and particles closest to the two coordinates are chosen as the output particles after each cycle. In other words, the association is not specifically between two chosen input particles and two chosen output particles but rather the particles nearest to the two output regions given the potential rearrangements and that we are training for direction of displacement and not magnitude (or displacement). (b) To retrieve the association in the form of a flow pattern, the black arrows denoting the direction of driving for the loading stage of the cycle and the grey arrows denote the direction of displacement (without driving). Simply put, only the input particles are driven in the retrieval stage. The magenta edges represent contacts gained during a training cycle and the dashed orange edges lost during a training cycle. Other contact edges are shown in lighter grey. (c) Schematic plot of the cosine similarity, or the cosine of the angle denoted in (b) as a function of the retrieval cycle, which is our performance measure.
• Figure 2: Easy task. (a) Opposite-direction training schematic: two fixed input particles (blue) are cyclically driven while two region-selected output particles (green) are driven in opposite directions (arrows show imposed displacements). (b) Same-direction training schematic: same inputs, but the two outputs are driven in the same direction during training. (c) Retrieval schematic: only the inputs are driven; at maximum input displacement we measure the instantaneous output displacement directions and define the angle $\theta$ between them. (d) Representative displacement field at maximum input displacement during retrieval for the untrained system (blue: inputs; green: outputs; gray: passive particles). (e) Cosine similarity versus retrieval-cycle number, computed as $\cos\theta$: black, untrained (direct retrieval); blue, opposite-direction training; green, same-direction training. (f) Total contact change $\mathrm{TCC}(t)$ measured at the cycle start during training, defined relative to the baseline contact network in the first cycle (see text for definition). (g) Total contact change $\mathrm{TCC}(t)$ measured at the cycle start during retrieval, again referenced to the same baseline contact network $\mathbf{C}_0$ from the first cycle.
• Figure 3: Hard task (opposite-direction training: opposite-direction output drive) (a) Retrieval schematic: only input particles are driven; snapshot at the halfway point of a retrieval cycle when the input particle reaches its maximum displacement; $\theta_1$ and $\theta_2$ are measured as the deviation angles between output-particle displacement directions and their target displacement directions. (b) Cosine similarity vs test cycles, computed from $\theta_1$ and $\theta_2$. Black curve shows the untrained case (no training, direct retrieval). Blue curve uses region driving rule. Green curve uses the same protocol with an output-relaxation step during training, in which every block of five cycles leaves the outputs undriven in the fifth cycle to allow relaxation. (c) Average energy versus cycle number. Each curve represents the mean across 100 independent runs. The vertical red dashed line separates the training phase (left) from the retrieval phase (right), with connecting lines showing the transition between phases. Data points correspond to the energy at the initial position and maximum displacement position within each cycle. (d) Total contact change, $\mathrm{TCC}(t)$, versus training cycle number for the trained protocols (blue/green), computed from the contact status at the cycle start relative to the baseline contact set at the first cycle start. (e) $\mathrm{TCC}(t)$ versus retrieval cycle number (same definition as in (d)). Error bars indicate variability across 100 independent realizations.
• Figure 4: Hard task (same-direction training: same-direction output drive) (a) Retrieval schematic: only input particles are driven; snapshot at the halfway point of a retrieval cycle when the input particle reaches its maximum displacement; $\theta_1$ and $\theta_2$ are measured as the deviation angles between output-particle displacement directions and their target displacement directions. (b) Cosine similarity vs retrieval cycles, computed from $\theta_1$ and $\theta_2$. Black curve shows the untrained case (no training, direct retrieval). Blue curve uses region driving of the outputs in every training cycle. Green curve uses the same protocol with an output-relaxation step during training, in which every block of five cycles leaves the outputs undriven in the fifth cycle to allow relaxation. (c) Total contact change, $\mathrm{TCC}(t)$, versus training cycle number for the trained protocols (blue/green), computed from the contact status at the cycle start relative to the baseline contact set at the first cycle start. (d) $\mathrm{TCC}(t)$ versus retrieval cycle number (same definition as in (c)). Error bars indicate variability across 100 independent realizations.
• Figure 5: Quadrupolar task. (a) Training protocol: two input particles (blue) are cyclically driven in opposite vertical directions, while two output particles (green) are cyclically driven in opposite horizontal directions (arrows indicate the imposed driving directions), forming a quadrupolar pattern. (b) Retrieval protocol: only the input particles are driven; output particles are undriven and respond mechanically. At maximum input displacement we measure the output displacement directions and define deviation angles $\theta_1$ and $\theta_2$ relative to the target directions (dotted arrows). (c) Same geometry as (a) but with enlarged input particles (used in the parameter variations reported in panel (d)). (d) Cosine similarity versus retrieval cycle number for the quadrupolar task. Black: untrained (direct retrieval without training). Blue: baseline training. Green: training with intermittent relaxation (every 5 training cycles the outputs are not driven). Purple: same as green but with an annealed output drive during training, where the output amplitude is ramped down linearly from $A_{\mathrm{out}}=4.2$ (first training cycle) to $A_{\mathrm{out}}=3.0$ (final training cycle), together with a reduced Lennard--Jones length prefactor ($\sigma_{\mathrm{coeff}}=0.95$ instead of $1.0$), which shortens the interaction length scale. Yellow: same as purple with increased input particle size (input size $=8.4$). (e) Total contact change $\mathrm{TCC}(t)$ versus training cycle number for the same protocols/labels as in (d). (f) $\mathrm{TCC}(t)$ versus retrieval cycle number for the same protocols/labels as in (d) (see text for definition of $\mathrm{TCC}$).Error bars indicate variability across 100 independent realizations.