Fig. 4: Learning noise-induced transitions in a bistable non-gradient system.
From: Learning noise-induced transitions by multi-scaling reservoir computing
a Schematic of transitions in the 2D bistable non-gradient system. b Generated time series from Eqs. (14) and (15) (a = b = 5, c = 0, ε1 = ε2 = 0.3, u1(0) = 0, u2(0) = 2, δt = 0.002) with t = 40 as the ground truth. c The trained slow-scale model transforms ten different start points into ten different slowly time-scale series (color lines), t ∈ [40, 80], and the noise distribution is separated in the training phase. d Result of prediction using the slow-scale model and the noise distribution in c. e The number of transitions for the 100 replicates simulated in t ∈ [40, 80] and that from the 100 predicted trajectories matches. f Histograms of transition time for the test and predicted data. The transition occurs when the time series crosses the zero point in the u1-direction without returning for 50δt. The transition time is defined as the interval between two consecutive zero crossings.
