A Supplemental Details

A.1 Data Generation A.1.1 Closure modeling For the present case, the initial condition is given by: u (x, 0) = The 2048 mesh point high-resolution solution is generated using the Fourier-Galerkin spectral method [41] with the 4th order Runge-Kutta method for time stepping. From the box-filtered initial condition, the 32-point low-resolution solution is conducted using central differencing for the spatial derivatives. This choice does not introduce additional artificial viscosity; thus, the solution without closure is naturally unstable. The high-resolution is computed at a small time-step, yet is down-sampled temporally at an interval equal to the low-resolution time step size t =0.0075 s. In this setting, u can be regarded as fully resolved, thus the numerical residual r ( u), defined in Eq. (10), is zero.