  Long-term stability is a critical property for deep learning-based
data-driven digital twins of the Earth system. Such data-driven digital twins
enable sub-seasonal and seasonal predictions of extreme environmental events,
probabilistic forecasts, that require a large number of ensemble members, and
computationally tractable high-resolution Earth system models where expensive
components of the models can be replaced with cheaper data-driven surrogates.
Owing to computational cost, physics-based digital twins, though long-term
stable, are intractable for real-time decision-making. Data-driven digital
twins offer a cheaper alternative to them and can provide real-time
predictions. However, such digital twins can only provide short-term forecasts
accurately since they become unstable when time-integrated beyond 20 days.
Currently, the cause of the instabilities is unknown, and the methods that are
used to improve their stability horizons are ad-hoc and lack rigorous theory.
In this paper, we reveal that the universal causal mechanism for these
instabilities in any turbulent flow is due to \textit{spectral bias} wherein,
\textit{any} deep learning architecture is biased to learn only the large-scale
dynamics and ignores the small scales completely. We further elucidate how
turbulence physics and the absence of convergence in deep learning-based
time-integrators amplify this bias leading to unstable error propagation.
Finally, using the quasigeostrophic flow and ECMWF Reanalysis data as test
cases, we bridge the gap between deep learning theory and fundamental numerical
analysis to propose one mitigative solution to such instabilities. We develop
long-term stable data-driven digital twins for the climate system and
demonstrate accurate short-term forecasts, and hundreds of years of long-term
stable time-integration with accurate mean and variability.
