variable $X$. For the direct numerical solution, the governing equations have to be known and the numerical resolution is dictated by the fast and small-scale variable $Z$, resulting in high computational costs. In the data-driven approach, the governing equations do not have to be known, $Y$ and $Z$ do not have to be known, and evolution of $X$ is predicted just from knowing the past observations of $X$, leading to low computational costs.[]{data-label="data_drive_demo"}](data_driven_model.png){width="\textwidth"}

Materials and Methods {#sec_method}
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The multi-scale Lorenz 96 system {#sec:RK4}
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### Numerical solution

We have used a 4th-order Runge-Kutta solver with time step $\Delta t=0.005$ to solve the system of Eqs. (\[eq:L1\])-(\[eq:L3\]). The system has been integrated for $100$ million time steps to generate a large dataset for training, testing, and examining the robustness of the results. In the Results section, we often report time in terms of model time units (MTUs), where 1 MTU $= 200\Delta t$. In terms of Lyapunov timescale, 1 MTU in this system is $\approx 4.5/\lambda_{max}$ [@thornes2017use], where $\lambda_{max}$ is the largest positive Lyapunov exponent. In terms of the $e$-folding decorrelation timescale ($\tau$) of the leading principal component (PC1) of $X$ [@khodkar2019reduced], we estimate 1 MTU $\approx 6.9 \tau_\mathrm{PC1}$. Finally, 