Extreme Events Prediction from Nonlocal Partial Information in a Spatiotemporally Chaotic Microcavity Laser

The forecasting of high-dimensional, spatiotemporal nonlinear systems has made tremendous progress with the advent of model-free machine learning techniques. However, in real systems it is not always possible to have all the information needed; only partial information is available for learning and forecasting. This can be due to insufficient temporal or spatial samplings, to inaccessible variables, or to noisy training data. Here, we show that it is nevertheless possible to forecast extreme event occurrences in incomplete experimental recordings from a spatiotemporally chaotic microcavity laser using reservoir computing. Selecting regions of maximum transfer entropy, we show that it is possible to get higher forecasting accuracy using nonlocal data vs local data, thus allowing greater warning times of at least twice the time horizon predicted from the nonlinear local Lyapunov exponent.

Figure 1

(a) Microlaser near-field image above lasing threshold. (b)–(e) Simultaneous recording of scaled intensities at $C$ (fixed photodetector, upper) and $M$ (mobile photodetector, lower) for specific events at $C$ (placed at $t=0$): an EE is present (b),(d),(e) or not present (c) at $C$. The photodetector is located at $x_M=0\mathrm{\mu }\mathrm{m}$ in (b), $x_{M_2}=+12.3\mathrm{\mu }\mathrm{m}$ in (d), and $x_{M_1}=-13.8\mathrm{\mu }\mathrm{m}$ in (e). Red and black lines: average timetraces over the ensemble of EE or NE and associated standard deviations (orange and light blue shaded areas). Dark red: largest EE’s timetrace, in (b),(d),(e) [respectively, single random NE in (c)] recorded at $C$ with the simultaneous timetrace recorded at $M$ (light blue). Red dashed line: EE threshold $2\times H_s$ (to be compared to the pulse height $H$ from the trough to peak, not to the peak value). Yellow shaded area in (e): information used for the prediction of an event at $t=0$, with history time ${\tau }_h$ and warning time ${\tau }_w$.

Figure 2

(a) Largest Lyapunov exponent ${\lambda }_M=\max (\lambda )$ and Kaplan-Yorke dimension $D_{KY}$ of the timetraces versus recording position. (b) Initial error growth rate for the laser intensity versus time computed at $C$ for both photodetector recordings. The extracted prediction time horizon (see text) is 0.47 ns.

Figure 3

Effective transfer entropy $T_{M\to C}^{\mathrm{eff}}$ as a function of $x_M$ and delay ${\tau }_w$ for ${\tau }_h=1,\mathrm{\Delta }t=0.050\mathrm{ns}$ and (b) at $M_2$ for ${\tau }_h=0.050$, 0.250, 0.500 ns. $P_1$ and $P_2$ locate the main precursor regions in the spatiotemporal diagram. $T_{M\to C}^{\mathrm{eff}}$ is smoothed by a small Gaussian kernel (see SM for the original data).

Figure 4

(a) Mean EE forecasting accuracy at $C$ with ${\tau }_h=1.75\mathrm{ns}$ versus the warning time ${\tau }_w$, using either local (at $C$) or nonlocal (at $M_1$ or $M_2$) data as input. The mean and standard deviation are computed with 10 different realizations of the reservoir. Thin line: nonlocal forecasting at $M_2$ using logistic regression alone. (b) Histogram of the testing dataset heights’ distribution (blue, NE; red, EE events). (c) Classification probability as an EE (green) or a NE (complementary, orange) of an event of actual height $H$ occurring ${\tau }_w=1\mathrm{ns}$ in the future at $C$ from the knowledge of a history of ${\tau }_h=1.75\mathrm{ns}$ duration of the nonlocal data at $M_2$. Two different realizations of the testing dataset are illustrated (plain and empty histograms).