Robustly optimal dynamics for active matter reservoir computing

TL;DR

We address how active matter can serve as a physical reservoir for time-series prediction by embedding chaotic driving and using coarse-grained observations to train a linear readout. The study identifies a robust near-critically damped regime, governed by intrinsic relaxation dynamics, that yields superior RC performance across multiple chaotic inputs and even with minimal single-particle substrates. The findings connect microscopic damping, velocity-fluctuation correlations, and interfacial dynamics to efficient information processing in nonequilibrium matter, offering interpretable insights beyond abstract neural networks. This work suggests practical pathways for in materio computation and guides future exploration of dissipative, many-body substrates for neuromorphic learning.

Abstract

Information processing abilities of active matter are studied in the reservoir computing (RC) paradigm to infer the future state of a chaotic signal. We uncover an exceptional regime of agent dynamics that has been overlooked previously. It appears robustly optimal for performance under many conditions, thus providing valuable insights into computation with physical systems more generally. The key to forming effective mechanisms for information processing appears in the system's intrinsic relaxation abilities. These are probed without actually enforcing a specific inference goal. The dynamical regime that achieves optimal computation is located just below a critical damping threshold, involving a relaxation with multiple stages, and is readable at the single-particle level. At the many-body level, it yields substrates robustly optimal for RC across varying physical parameters and inference tasks. A system in this regime exhibits a strong diversity of dynamic mechanisms under highly fluctuating driving forces. Correlations of agent dynamics can express a tight relationship between the responding system and the fluctuating forces driving it. As this model is interpretable in physical terms, it facilitates re-framing inquiries regarding learning and unconventional computing with a fresh rationale for many-body physics out of equilibrium.

Figures (50)

• Figure 1: Reservoir computing with active matter concept. The problem to solve is predicting the future state of a chaotic time series $\bm{Y}_{\text{target}}^{\text{train}}$ (the Lorenz-63 attractor, red trajectory) using reservoir computing. To this end, the time series is continuously injected into a swarm of active particles (blue) as a perturbing signal called the driver (red spiked ball, driver position $\bm{x}_d(t) \equiv \bm{Y}_{\text{target}}^{\text{train}}(t)$). This perturbation is modeled as a repulsive force $\bm{F}_d$ between the driver and the agents. A set of intrinsic interactions governs the swarm agents: They align their direction of motion ($\bm{F}_a$), avoid collisions ($\bm{F}_r$), seek a constant speed ($\bm{F}_{sc})$, return to the center of the simulation box ($\bm{F}_h$), and are limited in their overall response (force clamp) (see Sect. \ref{['sec:mm-interactions']}). The non-linear, non-equilibrium response of the swarm is then measured in a coarse-grained fashion (green observation kernels, see also Fig. \ref{['fig:gaussian_kernels_placement']}) and stored in a matrix $\mathrm{X}^{\text{train}}$ for each time step. Together with the original driver trajectory, this enables training a linear readout layer with weights $\bm{W}_{\text{out}}$. Using these weights, one can predict the future state $t + \Delta t$ at each time step $t$ of a new time series $\bm{Y}_{\text{target}}^{\text{test}}$. This new time series must follow the same dynamics as the one used for training, but may have different initial conditions. In this paper, we perform reservoir optimization: we aim to find the optimal set of agent interactions and understand the resulting agent dynamics in terms of underlying physics.
• Figure 2: A critically damped dynamical regime shows optimal predictive performance for active matter reservoir computing. (a) Varying the parameters of the speed-controlling force reveals regimes with stark differences in predictive performance, corresponding to different agent dynamics. The heatmap shows varied target agent speeds $s \in [10^{-5}; 10^{2}]$ that a particle will accelerate or decelerate towards (see Eq. \ref{['eq:speed-controller']}), and the strength coefficient $K_{sc} \in [10^{-5}; 10^{2}]$ of this force. In the center diagonal enclosed by the dashed yellow lines, there is a critically damped regime with suppressed agent excitations and a spatially well-confined collective swarm (circle, pyramid, square symbols). Moving towards lower speed-controller strengths and higher target agent speeds yields a response that appears much more disordered than coherent-collective (diamond, star symbols). Higher speed-controller strengths and lower target agent speeds lead to arrested particle motion (nabla symbol), which becomes overdamped when choosing a smaller integration time step (see Fig. \ref{['fig:Lymburn_critical_scans_speed_controller_smaller_integration_time_step']}). The hexagon/"L" denotes the optimal dynamical regime presented in Ref. Lymburn2021 (see Supplementary Video Tab. \ref{['tab:supplementary_videos_lymburn_reproductions']}/2 for a snapshot and visualization). (b) Snapshots of swarms in different dynamical regimes marked as symbols in sub-figure (a); color indicates agent speed. Refer to Tab. \ref{['tab:supplementary_videos_speed_controller']} for corresponding videos. (c) Predictive performances $P$ are higher in the critically damped regime (pyramid symbol, $P = 0.88$, $P_y = 0.79$, $\mathrm{NRMSE}= 0.54$, $\mathrm{NMSE}=0.29$, $\mathrm{sMAPE}= 6.4\,\%$ ) than in the underdamped regime (cross symbol, $P = 0.52$, $P_y = 0.40$, $\mathrm{NRMSE}= 0.88$, $\mathrm{NMSE}=0.78$, $\mathrm{sMAPE}= 12\,\%$ ). $P \equiv P_x$ is defined as the correlation coefficient specified in Eq. \ref{['eq:correlation-coefficient']} between the first dimension of the actual and the predicted time series. The gray curve shows the first dimension of the actual time series, while the colored curves show predictions $\Delta t = 0.5$ time units ahead made with the active matter reservoir at each time point. (d) Agent-averaged mean squared displacements (MSDs) after the Lyapunov time of the Lorenz-63 attractor $t^{\text{L63}}_{\text{lyap}} \approx 1.1$Viswanath1998 for different speed-controller parameter combinations. Low MSDs separate the underdamped from the near-critical (and arrested) regime, and correlate with high predictive performances. They are thus reliable proxies for consistent, controlled agent responses to changing external stimuli. Parameter values $(K_{sc}, s)$ for symbols: cross: (0.00005, 18.32981); diamond: (0.00379, 0.26367); square: (0.00886, 0.11288); pyramid: (0.02069, 0.04833); circle: (0.04833, 0.02069); nabla: (0.26367, 0.00379).
• Figure 3: Characterizing intrinsic microscopic relaxation dynamics (a,b) as well as dynamical correlations under driving (c,d). Symbols correspond to parameter combinations changing speed-controller settings $(K_{sc}, s)$ (and resulting damping dynamics) as shown in Fig. \ref{['fig:Lymburn_critical_speed_controller_scan_larger']}. (a,b) Intrinsic relaxation dynamics for a system without external driving in the absence of any collective effects (without agent-agent interactions, only subject to the homing force, the speed controller, and the sigmoid force wrapper described in Sect. \ref{['sec:mm-interactions']}), the time step used is $\Delta t = 0.02$. The system relaxes from an initial, uniformly random distribution to a steady state, analogous to a damped harmonic oscillator. (a) Structural relaxation of a population of non-interacting agents measured as the mean radial distance to the center of the simulation box, normalized by the initial radial distance at time $t_0 = 0$. Sustained oscillations around the center of the simulation box (underdamped regime, cross symbol) become suppressed when moving towards higher speed-controller strengths $K_{sc}$ and lower target agent speeds $s$ (diamond symbol). Critical damping is situated between the square and pyramid symbols, at the threshold where a single oscillation disappears during the initial phase of the relaxation. (b) Corresponding dynamical relaxation: the relative deviation of agent speed from the set target agent speed $s$. Error bands indicate the standard error of the mean over $N=200$ agent trajectories. For reference, the Lyapunov timescale of the Lorenz-63 driver is shown in the time plots (a)-(c), i.e., $t^{\text{L63}}_{\text{lyap}} \approx 1.1$Viswanath1998 as a gray dashed line. (c,d) Dynamical correlations under driving for driven agents subject to the full active matter model described in Sect. \ref{['sec:mm-interactions']}. (c) The absolute value of the velocity auto-correlations for selected damping dynamics. The optimal dynamics (around pyramid) entail unique features described in the main text. (d) The connected velocity correlation function (CVC) for different parameter combinations. The optimally damped regime shows the strongest connected velocity fluctuation correlations and anti-correlations, measured spatially via a radial distance around each particle.
• Figure 4: Phenomenology of optimally damped soft matter systems under driving. Displayed are the response dynamics of $N=500$ agents under quasi-stationary and abrupt driving, which are both part of Lorenz-63 dynamics. Upper panel: Under adiabatic, quasi-stationary driving, agents form a stable interface around the driver, established by the global attractive force to the center of the simulation box (homing force) and the local repulsive force from the driver. Lower panel: Abrupt driving causes the agent-driver interface to break. The driver dashes through the swarm, causing local shear thinning. A new interface forms once the driver slows down. Self-healing through viscous backflow converts the former interface to bulk. The driver tail length corresponds to five integration time steps ($5 \Delta t = 0.1$), which is also the time between each snapshot. The snapshots were taken at times $t_i = \{1.0, \dots, 1.4\}$ (quasi-stationary response) and $t_i = \{3.0, \dots, 3.4\}$ (highly non-linear response); the Lorenz-63 driving protocol is the same as shown in Fig. \ref{['fig:Lymburn_critical_speed_controller_scan_larger']}c. Simulation parameters correspond to those marked by the pyramid symbol in Fig. \ref{['fig:Lymburn_critical_speed_controller_scan_larger']}a ($K_{sc} = 0.02069$, $s = 0.04833$). Refer to Supplementary Video Tab. \ref{['tab:supplementary_video_phenomenology']}/1 for the corresponding video.
• Figure 5: The near-critically damped regime shows strong correlations and anti-correlations of agent velocity fluctuations.. (a) The dynamical susceptibility $\chi$ (Eq. \ref{['eq:dynsus']}) captures the intensity and spatial reach between velocity fluctuation correlations (Eq. \ref{['eq:velocity-fluctuation']}, deviations of individual agent velocities from the system's center of mass velocity). The high predictive performance regime found in Fig. \ref{['fig:Lymburn_critical_speed_controller_scan_larger']}a correlates with the $\chi > 5.0$ region. (b) The anti-correlative dynamical response $\text{CVC}(r_{\text{min}})$, the minimum of the connected velocity correlation function (Eq. \ref{['eq:connected_velocity_correlation']}) shown in Fig. \ref{['fig:damping']}d), quantifies the maximum extent of anti-correlations between agent velocity fluctuations and correlates with the predictive performance Fig. \ref{['fig:Lymburn_critical_speed_controller_scan_larger']}a. The strongest signal is observed just below critical damping at the pyramid symbol. Extrema around the "L" symbol could stem from strong rotational modes that were not considered for velocity fluctuation computations. (c) Visualizations of velocity fluctuations for different parameter combinations (symbols) in the speed-controller parameter scans of (a,b). Arrow length indicates strength, and color indicates orientation of velocity fluctuations. Refer to Tab. \ref{['tab:supplementary_videos_speed_controller_velocity_fluctuations']} for corresponding videos.