Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable

Figure 1.  Inference of a time-series of the Reynolds number of a fluid flow. Time-series of $ s_1 = \tilde{R}_{\lambda} $ is inferred from the reservoir model in comparison with that of a reference data obtained by the direct numerical simulation of the Navier–Stokes equation (top left). The variable $ t^\prime \; ( = t-T>0) $ denotes the time after finishing the training phase at $ t = T $. The inference errors $ \varepsilon_1, \varepsilon_2 $ defined by $ \varepsilon_1(t) = | \mathbf{s}(t)-\hat{ \mathbf{s}}(t)| $, and $ \varepsilon_2(t) = |s_1(t)-\hat{s}_1(t)| = |\tilde{R}_{\lambda}(t)-\hat{\tilde{R}}_{\lambda}(t)| $ are shown to increase exponentially due to the chaotic property (top right). In the bottom figure switching between laminar state with a small amplitude fluctuation and bursting state with a large amplitude fluctuation appear in an inferred time-series of $ s_1 = \tilde{R}_{\lambda} $, which are observed in the actual time-series

Figure 2.  Reproducing the delay property which is to be satisfied for the successfully inferred time-series $ \hat{\mathbf s} $. We observe that for all values of $ m = 2, \cdots, 14 $ and for most $ t^{\prime} $, $ \hat{s}_1(t^\prime)\approx\hat{s}_{m}(t^\prime+(m-1)\Delta\tau) $, although the time-series of only $ \hat{s}_1(t^\prime) $ and $ \hat{s}_{14}(t^\prime+13\Delta\tau) $ $ (7000\le t^{\prime}\le 8000) $ are shown

Figure 3.  Poincaré points on the plane $ (s_2, s_3) $ along the trajectory $ \hat{ \mathbf{s}} $ obtained from the reservoir model (red) and $ \mathbf{s} $ from the Navier-Stokes equation (blue). The time length of each trajectory is $ 90000 $. The Poincaré section is defined by $ {s}_{1} = 0, k_ge d{s}_{1}/dt>0 $. Two sections are similar to each other, although a trajectory generated from the reservoir model does not cover some region of bursting states

Figure 4.  Density distributions generated from trajectories for a variable $ s_1 $ obtained from the constructed reservoir model (reservoir output) and from the direct numerical simulation of the Navier-Stokes equation (actual). Each trajectory with a time-length 50000 has a different initial condition. The distributions are similar to each other in the sense that the peak is taken at $ s_1\approx0.2 $, and the distribution has relatively long tails

Figure 5.  Inference of a time-series of the Reynolds number for $ t^{\prime}>T_{\text{out}} $ ($ T_{\text{out}} = 1000 $) using the reservoir model constructed by using the training data for $ t^\prime\le 0 $ (see Fig. 1). We use the same $ \mathbf{W}_\text{in}, \mathbf{A}, \mathbf{W}^*_\text{out} $ and $ \mathbf{c}^* $ as those used for the model inferring the trajectory in Fig. 1. But we use the time-series $ s_1(t^\prime) $ for $ T_{\text{out}}- T_1<t^\prime<T_{\text{out}} $ as an initial condition, where $ T_1 $ is the transient time for the reservoir state vector $ \mathbf{r}(t) $ to be converged. In the top panel, switching between laminar and bursting states is observed in the inferred trajectory. The bottom panel is the enlargement of the top panel, and shows that the model has a predictability for $ 1000<t^\prime<1080 $

Figure 6.  Inference of time-series of the Reynolds number in many time-intervals $ T_{\text{out}}<t^{\prime}<T_{\text{out}}+250 $ ($ T_{\text{out}} = 500, 1000, \cdots, 6000 $) using the same reservoir model constructed by using the training data for $ t^\prime\le 0 $ (see Fig. 1 and 5.) As in Fig. 5, we only change the initial condition for each case, while the model is fixed after the appropriate choice of $ \mathbf{W}_\text{in}, \mathbf{A}, \mathbf{W}^*_\text{out} $ and $ \mathbf{c}^* $ is determined by using the training data for $ t^{\prime}<0 $

Figure 7.  Auto-correlation function $ C(x) $ for a trajectory $ \{R_{\lambda}(t)\} $ with respect to the value of time-delay $ x $ (left), and its enlarged figure (right). Auto-correlation function $ C(x) $ is shown together with the straight lines $ \pm0.3, \pm0.5 $ (left panel), and $ 0.3, 0.7 $ (right panel). Each of the different colors represents $ C(x) $ computed from a trajectory from a different initial condition with time-lengths 5000. The difference is mainly due to the intermittent property of the dynamics. In the left panel the envelope $ C_e(x)( = \exp(-x/60)) $ is shown to go below 0.5 when $ x \approx 40 $, and also go below 0.3 when $ x \approx 75 $. From the right panel $ C(x) $ is shown to go below $ 0.7 $ at the first time, when $ x \approx 3.0 $, and go below $ 0.3 $ at the first time, when $ x \approx 5.0 $