is related to the time step of the numerical solver $(\Delta t)$ and largest positive Lyapunov exponent $(\lambda_{max})$ as $1$ MTU $=$ $200 \Delta t \approx 4.5/\lambda_{max}$ (see section \[sec:RK4\]). Note that here we are presenting raw data, which have not been standardized for prediction/testing yet.[]{data-label="xyz_demo"}](xyz_demo.eps){width="\textwidth"}

*Our objective is to predict the spatio-temporal evolution of $X(t)$ using a data-driven model that is trained on past observations of $X(t)$* (Figure \[data\_drive\_demo\]). In the conventional approach of solving Eqs. (\[eq:L1\])-(\[eq:L3\]) numerically, the governing equations have to be known, initial conditions for $Y(t)$ and $Z(t)$ have to be available, and the numerical resolution is dictated by the fast/small-scale variable $Z$, leading to high computational costs. In the fully data-driven approach that is our objective here, the governing equations do not have to be known, $Y(t)$ and $Z(t)$ do not have to be observed at any time, and evolution of $X(t)$ is predicted just from knowledge of the past observations of $X$, leading to low computational costs. To successfully achieve this objective, a data-driven method should be able to

1.  Accurately predict the evolution of a chaotic system along a trajectory for some time,

2.  Account for the effects of $Y(t)$ and $Z(t)$ on 