Figure 1

Circles: standard probability distribution function (PDF) $P_L\left(p_c\right)$ as a function of the percolation threshold $p_c$ (in percent) for $L=20$. Crosses: conditional PDF $P_L(p_c\mid p=0.5)$ conditioned on those systems which have not percolated for a fixed occupation density $p=0.5$. Dots: conditional PDF $P_L(p_c\mid p=0.53)$ conditioned on those systems which have not percolated for a fixed occupation density $p=0.53$. Squares: conditional PDF $P_L(p_c\mid p=0.55)$ conditioned on those systems which have not percolated for a fixed occupation density $p=0.55$.Reuse & Permissions

Figure 2

PDF $P_L(p_c\mid p,p_{\xi })$ as a function of $p_c$ (in percent) conditioned on both $p$ and $p_{\xi }=6\%$, where $p_{\xi }$ is the fraction of sites belonging to the largest cluster, for different values of $p$ (crosses, $p=0.4$; dots, $p=0.45$; squares, $p=0.5$). For comparison, the unconditional distribution of the first prediction level is also shown with open circles.Reuse & Permissions

Figure 3

PDF $P_L(p_c\mid p,\xi )$ as a function of $p_c$ (in percent) conditioned on both $p$ and $\xi =0.2L$, where $\xi$ is the largest of the linear size projected on the $x$ and $y$ axes of the largest cluster within the system, for different values of $p$ (dots, $p=0.35$; squares, $p=0.4$). For comparison, the unconditional distribution of the first prediction level is also shown with open circles.Reuse & Permissions

Figure 4

Same as Fig. 3 for a fixed $p=40\%$ and different values of $\xi$: $\xi ∕L=0.04$ (crosses), $\xi ∕L=0.06$ (dots), $\xi ∕L=0.08$ (squares), and $\xi ∕L=0.1$ (triangles). The open circles represent the unconditional PDF $P_L\left(p_c\right)$ for reference.Reuse & Permissions

Figure 5

Panel (a): relative specific information gain $I(p,p_{\xi })$ as a function of $p$ (in percent) for various values of $p_{\xi }=2\%,4\%,6\%,8\%,10\%$ from left to right. Panel (b): relative specific information gain $I(p,\xi )$ as a function of $p$ for various values of $\xi ∕L=10\%,20\%,30\%,40\%,50\%,60\%$ from left to right.Reuse & Permissions

Figure 6

rms $Q(p,p_{\xi })$ (in percent) of the prediction errors defined by Eq. (4) with Eq. (5), for the quantiles $q=5\%$, $q=50\%$ and $q=95\%$ of the distribution of $\xi$ at fixed $p$, when using $P\mathbf{(}{p}_c\mid p,{\xi }^q\left(p\right)\mathbf{)}$ as the predictor, as a function of the damage parameter $p$ (in percent). Triangle, $q=5\%$; dots, $q=95\%$; crosses, $q=50\%$; circles, $Q\left(p\right)$ obtained using the conditioning only on $p$. This rms $Q(p,p_{\xi })$ should be compared with the standard deviation equal to $4.66\%$ of the unconditional distribution of percolation thresholds, to illustrate the gain in prediction accuracy deriving from the added information.Reuse & Permissions

Figure 7

Gain in rms $Q\left(p\right)-Q(p,\xi )$ when adding the information on $\xi$, where $Q(p,\xi )$ is shown as the triangles $(q=5\%)$, dots $(q=95\%)$, and crosses $(q=50\%)$ and $Q\left(p\right)$ is shown in Fig. 6 with circles.Reuse & Permissions

Figure 8

A specific realization of the space-time evolution of fiber damage for a system of $2^8=256$ fibers. The fibers are numbered sequentially from 1 to 256 along the vertical axis. When a given fiber $i$ breaks at some time $t_i$, a symbol + represents the spatial position and failure time of this event.Reuse & Permissions

Figure 9

(Top) Two different measures of the cumulative number of broken fibers as a function of time $t$ for a given realization. The thick curve shows the unconditional cumulative number of broken fibers. The thin curve shows the (conditional) cumulative number of broken fibers, which broke either within the largest crack identified up to time $t$ or within its complement in their pair within the hierarchy. (Bottom) This graph shows the same two curves in log-log scales with time $t$ replaced by $t_c-t$, where $t_c$ is the global time of failure (only known at the end). This log-log representation allows us to visualize the power-law acceleration characterizing the final critical regime before complete rupture, which is much more apparent in the conditional cumulative number of broken fibers.Reuse & Permissions

Figure 10

Root mean square $Q\left(t\right)$ of the error or difference between predictions made at time $t$ of the global rupture time and the true realized one as a function of time $t$ for four distinct prediction schemes using different conditioning, similarly to Fig. 6 previously constructed for the percolation model. The system used here has $2^8$ fibers and $\rho =2$. The 엯 symbols correspond to a prediction at time $t$ of the failure time $t_c$ based solely on the information that the system has not yet broken. For the other curves, we constructed the distribution of failure times over ${10}^6$ realizations for the different conditioning. The triangles correspond to the rms $Q\left(t\right)$ obtained by using the 5% quantile of the distribution of failure times over these ${10}^6$ simulations. Specifically, for a given system and at a given time $t$, we measure the size $\xi$ of the largest failed cluster and then read from the distribution of failure times for the same time $t$ and same cluster size $\xi$ the 5% quantile that we take as the prediction for the failure time. Similarly for the crosses and dots corresponding, respectively, to the 50% and 95% quantiles. Note that in our system of $2^8$ fibers, there are eight possible sizes of “cracks” larger than 1—namely, 2,4,8,…,128,256. These curves are obtained by averaging over ${10}^5$ realizations. These rms $Q\left(p\right)$ for the five prediction schemes should be compared with the standard deviation equal to 0.0311 of the unconditional distribution of failure times $t_c$, to illustrate the gain in prediction accuracy deriving from the added information obtained from conditioning.Reuse & Permissions