Stable reproducibility of turbulence dynamics by machine learning

We have constructed a machine learning-based turbulence model for a shell model. Without $a\phantom{\rule{0}{0ex}}d$ $h\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}c$ stabilizations, the constructed model becomes unstable when the cutoff wavenumber is lower than $0.2{\ensuremath{\eta}}^{\ensuremath{-}1}$, where $\ensuremath{\eta}$ is the Kolmogorov length. This wavenumber characterizes the subordination of smaller to larger scales in turbulence. This finding highlights a theoretical limitation of turbulence modeling without stabilization techniques.