Machine-learning inference of fluid variables from data using reservoir computing

We infer both microscopic and macroscopic behaviors of a three-dimensional chaotic fluid flow using reservoir computing. In our procedure of the inference, we assume no prior knowledge of a physical process of a fluid flow except that its behavior is complex but deterministic. We present two ways of inference of the complex behavior: the first, called partial inference, requires continued knowledge of partial time-series data during the inference as well as past time-series data, while the second, called full inference, requires only past time-series data as training data. For the first case, we are able to infer long-time motion of microscopic fluid variables. For the second case, we show that the reservoir dynamics constructed from only past data of energy functions can infer the future behavior of energy functions and reproduce the energy spectrum. It is also shown that we can infer time-series data from only one measurement by using the delay coordinates. This implies that the obtained reservoir systems constructed without the knowledge of microscopic data are equivalent to the dynamical systems describing the macroscopic behavior of energy functions.

Figure 1

Partial inference of time series of microscopic variables in Fourier space of a fluid flow. Fourier variables ${\tilde{a}}_{{\eta }_1=(1,3,3,3)}$ (top) and ${\tilde{a}}_{{\eta }_2=(1,2,3,4)}$ (bottom) are inferred by using measured variables ${\tilde{a}}_{\eta }$ for $\eta \in S$ as well as the past time-series data for all the measured variables ${\tilde{a}}_{\eta }$ for $\eta \in S\cup \{{\eta }_1,{\eta }_2\}$. We can observe that the inferred time series almost coincide with the actual ones obtained by the direct numerical simulation of the Navier-Stokes equation even after a sufficiently long time has passed since the training phase finished. The inference errors in $l^1$-norm averaged over $t-T\in [0,2000]$ are $1.8\%$ and $3.5\%$ for ${\tilde{a}}_{{\eta }_1}$ and ${\tilde{a}}_{{\eta }_2}$, respectively.

Figure 2

Full inference of time series of macroscopic variables of a fluid flow. Time series of energy function $\tilde{E}(k,t)$ for $k=4$ (top) and 9 (middle) are inferred from the reservoir system in comparison with that of reference data obtained by direct numerical simulation of the Navier-Stokes equation. The inference error defined by $\varepsilon \left(t\right)={\sum }_{k=1}^{N_0}|\tilde{E}(k,t)-\widehat{\tilde{E}}(k,t)|/N_0(N_0=9)$ is shown to grow exponentially with time up to $t-T=100$ (bottom), which is inevitable for a chaotic behavior of a fluid flow. The growth of error within a short time depends a great deal on the direction of the perturbation vector $\{\tilde{E}(·,T+\mathrm{\Delta }t)-\widehat{\tilde{E}}(·,T+\mathrm{\Delta }t)\}$, and its slope can vary in different settings.

Figure 3

Energy spectrum $\overline{E}\left(k\right)$ reproduced from the reservoir computing. The spectrum is obtained from the full inference of an energy function $E(k,t)$, which is compared with that for reference data obtained by the direct numerical simulation of the Navier-Stokes equation. The coincidence of the two energy spectra implies that the reservoir system captures the dynamics of a fluid flow in a statistical sense, even after the time-series inference has failed due to the chaotic property (see Fig. 2). The Kolmogorov $-5/3$ law of the energy spectrum is shown as a reference. The relative error of the inferred variable $\overline{\widehat{E}}\left(k\right)$ from $\overline{E}\left(k\right)(k=1,\cdots ,9)$ is up to $1.3\%$.

Figure 4

Full inference of a macroscopic variable using the delay coordinates of only one measurement. We infer an energy function $\tilde{E}(4,t)$ for the same time range as in Fig. 2 (top) from only one measurement $\tilde{E}(4,t)$. The inferred time series of $\tilde{E}(4,t)$ is shown together with reference data obtained by the direct numerical simulation of the Navier-Stokes equation (top). Errors for the inference ${\varepsilon }_1\left(t\right)={\sum }_{n=0}^{35}{|\tilde{E}(4,t-n\mathrm{\Delta }\tau )-\widehat{\tilde{E}}(4,t-n\mathrm{\Delta }\tau )|}^2/36$ and ${\varepsilon }_2\left(t\right)=|\tilde{E}(4,t)-\widehat{\tilde{E}}(4,t)|$ are shown (bottom).