Fig. 3: Predicting the accurate transition time for a bistable gradient system with colored noise.

The system is the same as Eq. (10)⁴⁰. a The flowchart of predicting stochastic transitions with colored noise. The process for obtaining noise ζ_t follows that in Fig. 1a, and a second reservoir takes into ζ_t through matrix \({W}_{in}^{*}\) and has reservoir states \({{{{\bf{r}}}}}_{t}^{*}\) with a connection matrix A^*. The output matrix \({W}_{out}^{*}\) is trained to learn noise. b Target time series (x(0) = y(0) = z(0) = 1, b = 1, c = 0, ψ = 0.08, ϵ = 0.5, u₁(0) = − 1.5, δt = 0.01) with 8000δt, where a noise-induced transition occurs in t ∈ [22, 25] marked by the green dashed line. The noisy data from 580δt (with a range of 550δt to 650δt empirically suitable) before the stochastic transition at t = 22 is applied to predict the noisy time series in t ∈ [22, 25]. c The trained slow-scale model transforms ten different start points into ten different slowly time-scale series (color lines). d By repeating the process in a with the same hyperparameters, 50 predicted u₁(t) are obtained (fainter lines). The averaged predicted time series (thick green) matches the test data (coral). e Absolute error of the predicted 50 time series and its mean value (thick green).