Reservoir computing from collective dynamics of active colloidal oscillators

Abstract

Physical reservoir computing is a computational framework that offers an energy- and computation-efficient alternative to conventional training of neural networks. In reservoir computing, input signals are mapped into the high-dimensional dynamics of a nonlinear system, and only a simple readout layer is trained. In most physical implementations, the interactions that give rise to the dynamics cannot be tuned directly and high dimensionality is typically achieved through time-multiplexing, which can limit flexibility and efficiency. Here we introduce a reservoir composed of hundreds of hydrodynamically coupled active colloidal oscillators forming a fully parallel physical reservoir and whose coupling strength and fading-memory time can be tuned in situ. The collective dynamics of the active oscillators allow accurate predictions of chaotic time series from single reservoir readouts without time-multiplexing. We further demonstrate real-time detection of subtle hidden anomalies that preserve all instantaneous statistical properties of the signal yet disrupt its underlying temporal correlations. These results establish interacting active colloids as a reconfigurable platform for physical computation and edge-integrated intelligent sensing for model-free detection of irregularities in complex time signals.

Figures (8)

• Figure 1: Active colloidal oscillators.a A colloidal particle (grey) with position $\mathbf{r}_i$ is activated by a focused laser beam (green), inducing self-propulsion with velocity $\mathbf{v}_i$. b Active colloid steered toward a target position $\mathbf{q}_i$ (red). The propulsion direction $\mathbf{v}$ (solid arrow) deviates from the target direction (dashed line) due to a time delay between particle detection and laser positioning. c This misalignment induces an orbiting motion of the active colloid around the target position (blue arrow). d Experimental snapshot of 400 ACOs with target positions arranged on a hexagonal lattice of spacing $L$. e,f Trajectories (blue) of a single particle for damping parameters $\Gamma$ = 0 and $1.5\mathrm{\mu m}$. The gray area corresponds to the optical image of the particle (dashed circle). g,h Section of the oscillator array shown in D, with ACOs colored by their angular momentum $h_i$ for $\Gamma = 0$ and $1.5\mathrm{\mu m}$. i Experimentally measured probability distributions of $\left| h_i \right| = \left| \mathbf{v}_i \times (\mathbf{r}_i - \mathbf{q}_i) \right|$ for different values of $\Gamma$.
• Figure 2: Response of reservoir of 400 ACOs to an input signal.a-d Relaxation of the mean absolute velocity $\langle v \rangle$ (blue) after excitation with a $4\,\mathrm{\tau_{\text{d}}}$ square pulse for different parameters (A: $\Gamma = 1.4\,\mathrm{\mu m}, L = 12\,\mathrm{\mu m}$, B: $\Gamma = 1.4\,\mathrm{\mu m}, L = 30\,\mathrm{\mu m}$, C: $\Gamma = 0\,\mathrm{\mu m}, L = 12\,\mathrm{\mu m}$, D: $\Gamma = 0\,\mathrm{\mu m}, L = 30\,\mathrm{\mu m}$). The black and red curves are the running variance $\sigma^2_{10}[\langle v \rangle]$ with a window length of 10 $\tau_{\text{d}}$ and an exponential fit to $\sigma^2_{10}[\langle v \rangle]$, respectively. e Color map of the measured relaxation times $\tau_{r}$ for different combinations of $\Gamma$ and $L$. Orange symbols indicate the parameter combinations illustrated in A-D. See Fig. S1 for corresponding simulation results.
• Figure 3: Reservoir computing with arrays of coupled ACOs.a Coupling of the input signal (red) to the active colloidal oscillators (ACOs). The signal modulates each oscillator’s target position $\mathbf{q}$, which is translated along a line with a fixed but randomly chosen orientation. b Working principle of the reservoir. The signal $\mathbf{u}(t)$ drives all ACOs within the reservoir which is divided into $N$ rows with a one-step delay between rows, which propagates the signal through the ACO array. From the full reservoir state $\mathbf{s}(t) = \left\{ \mathbf{r}_1(t), \mathbf{v}_1(t), ... \, \mathbf{r}_i(t), \mathbf{v}_i(t) \right\}$, we construct readout vectors $\mathbf{x}(t)$ using Gaussian kernels (blue; see Methods). One linear readout layer $\mathbb{W}_\mathrm{out}$ produces the output $\mathbf{o}(t)$ (e.g. a prediction of the signal $\mathbf{u}(t + \Delta t)$). c Representative segment of a chaotic Mackey–Glass (MG) signal $\mathbf{u}(t + \Delta t)$ (red) together with output $\mathbf{o}(t)$ (blue) of reservoirs trained to predict the signal with forecasting horizons $\Delta t = 1$ (top) and $\Delta t = 10$ (bottom) from the validation split (i.e. not part of the training data) from experiments. d Heat map of the average prediction performance $N\!R\!M\!S\!E$ for $\Delta t = 10$ MG forecasting ($N = 10$) in simulations as a function of ACO spacing $L$ and damping parameter $\Gamma$. e Average prediction performance $N\!R\!M\!S\!E$ for ten step MG forecasting in simulations as a function of ACO separation $L$ in simulations ($N = 10$) for different prediction horizons $\Delta t$. f Change of average error $N\!R\!M\!S\!E$ for ten step MG forecasting with increasing number of injected signal states $N$ for different ACO separations $L$ in simulation. See Fig. S3 for a larger range of $N$. g Average error for ten step MG forecasting as a function of the fraction of driven ACOs $f_{\mathrm{driver}}$ and for different $L$ in simulations ($N = 10$).
• Figure 4: Detecting hidden anomalies via prediction errors.a Introduction of spiking anomalies to the Mackey–Glass (MG) series (black) by adding a sparse spiking signal (green), resulting in local outliers (red; spikes highlighted by green circles). b Top: MG signal $\mathbf{u}$ containing spiking anomalies (red) together with reservoir predictions $\mathbf{o}$ (light blue). Bottom: Squared prediction error $\left(\mathbf{o} - \mathbf{u}\right)^2$ over time, showing pronounced peaks at the anomaly locations (vertical lines). c Introduction of hidden anomalies by replacing short signal segments with intervals matching both instantaneous value and slope, preserving the visible continuity of the time series while altering its hidden memory structure. d Top: MG signal with hidden anomalies (red; highlighted by green circles) and corresponding predictions $\mathbf{o}$ (light blue). Bottom: Squared prediction error $\left(\mathbf{o} - \mathbf{u}\right)^2$, revealing clear peaks at the hidden anomaly times (vertical lines).
• Figure 5: Relaxation of the reservoir in simulations with arrays of 400 ACOs after excitation by displacing all target positions in the x-direction by $4\,\mathrm{\mu m}$ for $4\,\mathrm{\tau_d}$.