Long-term instabilities of deep learning-based digital twins of the climate system: The cause and a solution

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.