Data-driven prediction and prevention of extreme events in a spatially extended excitable system

Extreme events occur in many spatially extended dynamical systems, often devastatingly affecting human life, which makes their reliable prediction and efficient prevention highly desirable. We study the prediction and prevention of extreme events in a spatially extended system, a system of coupled FitzHugh-Nagumo units, in which extreme events occur in a spatially and temporally irregular way. Mimicking typical constraints faced in field studies, we assume not to know the governing equations of motion and to be able to observe only a subset of all phase-space variables for a limited period of time. Based on reconstructing the local dynamics from data and despite being challenged by the rareness of events, we are able to predict extreme events remarkably well. With small, rare, and spatiotemporally localized perturbations which are guided by our predictions, we are able to completely suppress extreme events in this system.

Figure 1

(b) Exemplary evolution of the dynamical variables $x_i$ (color coded) for all 100 units as well as (a) the evolution of the corresponding average value $\overline{x}\left(t\right)=N^{-1}{\sum }_{i=1}^N{x}_i\left(t\right)$. Panels (c), (d), and (e) show enlarged segments of panel (b).

Figure 2

Histogram of interevent intervals $t_{\text{IEI}}$ obtained for a realization of the system as defined in Sec. 2 (observation time of the data: ${10}^8$ time units). The rate of extreme events was estimated from the data as $r=1.54\times {10}^{-4}$ (extreme events were required to be separated by at least 150 time units to qualify as separate events). The dashed line represents a multiple of $exp(-r{t}_{\text{IEI}})$.

Figure 3

Scheme demonstrating the construction of an exemplary embedding vector ${\mathbf{s}}_i^t$ (blue [gray] box) for unit $i$ at time step $t$. The components of each embedding vector of unit $i$ are values of the multivariate time series $\mathbf{x}$. These values reflect the present and immediate past of unit $i$ ($t,...,t-m_t$) as well as the corresponding time series values of those units $j$ which are spatially close to unit $i$ ($j\in \{i-m_s,...,i-1,i+1,...,i+m_s\}$ if $i$ is not close to a boundary). The local embedding space of unit $i$ is populated with vectors ${\mathbf{s}}_i^T,...,{\mathbf{s}}_i^{m_t+1}$ before the future value $x_i^{t+1}$ is predicted via nearest-neighbor search. The embedding parameters, i.e., the numbers of spatial neighbors $m_s$ and delayed values $m_t$, are determined by minimizing the prediction error (see text).

Figure 4

(a) Temporal evolution of the first dynamical variables $x_i$ (color coded) of all units before and during two exemplary extreme events. Panels (b)–(e): Predictions which consist of 200 iterative steps and start at different points in time (indicated by vertical black lines).

Figure 5

(a) Three-dimensional projection of a 15-dimensional local embedding space of unit 40 constructed from time series of the training set (which included 13 extreme events). (1) marks a small subspace in which all vectors representing low-amplitude oscillations reside. Vectors corresponding to excitations (i.e., extreme events) propagating over unit 40 from lower-numbered units are colored blue (dark gray), and those from higher-numbered units are colored green (light gray). (b) Enlargement of region (2) of panel (a) showing vectors of the training set (green [light gray]) as well as an iterative prediction (blue [dark gray]) representing an excitation of unit 40. Each vector from the last prediction step is connected to its nearest neighbor (green [gray] triangle) with a dashed line. The image of the nearest neighbor as well as the vectors of the neighboring units are used to construct the next predicted vector. Colored arrows in both panels indicate the direction of temporal evolution. The axes $z_1, z_2$, and $z_3$ span the space of the three-dimensional projection as determined by isomap [37, 38], using ${\epsilon }_{\text{NG}}=0.5$ for the construction of the neighborhood graph.

Figure 6

(a) Dependence of the true-positive rate on the lead time $t_{\text{lead}}$ determined from all extreme events in the test set (TPR, dotted blue [dark gray] line, 66 events), determined only from the extreme events with onsets at the border of the chainlike topology (${\text{TPR}}_{\text{b}}$, red [gray], 27 events), and from those with onsets everywhere else (${\text{TPR}}_{\text{m}}$, green [light gray], 39 events). TPR values were obtained using ${\epsilon }_t=3$ and ${\epsilon }_s=5$. (b) Dependence of the false-positive rate (FPR) on the number of iterative prediction steps $t_{\text{step}}$.

Figure 7

Minimum amplitude (color coded, capped at ${10}^3$) of a perturbation at time $t=t_{\text{p}}$ of the inhibitory variable of unit $i$ to prevent an exemplary extreme event from occurring before $t=98$. The onset of the extreme event in the case of unperturbed dynamics was located at $t=0$ and $i=56$ (indicated by a red [gray] circle). At $t=1$, a second unit, $i=51$ (red [gray] circle), crossed the threshold.

Figure 8

(a) Temporal evolution of the mean value $\overline{x}$ of the voltage-like variables $x_i, i\in \{1,...,N\}$ for a control experiment spanning 204 000 time units. The gray background marks the region for which the temporal evolution of all $x_i$ is shown in panel (b). (b) Temporal evolution of $x_i$ [color coded as in Fig. 1] for all units $i$ and for an exemplary excerpt of panel (a) (marked gray). Blue (dark gray) circles mark points in time and in space for which prediction attempts correctly predicted the occurrence of an extreme event. Red (gray) triangles mark points in time and in space for which prediction attempts erroneously predicted the occurrence of an extreme event.