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.

翻译：长期稳定性是基于深度学习的数据驱动地球系统数字孪生的关键属性。此类数据驱动数字孪生能够实现极端环境事件的次季节与季节预测、需要大量集合成员的概率预报，以及用低成本数据驱动替代模型替换昂贵组件的计算可行的高分辨率地球系统模型。受计算成本限制，基于物理的数字孪生虽能长期稳定，却难以支持实时决策。数据驱动数字孪生以其低成本优势成为替代方案，可实现实时预测。然而，这类数字孪生仅能提供准确的短期预报——当时间积分超过20天后即出现不稳定性。目前，不稳定的成因尚不明确，用于提升稳定性的方法多属经验性处理，缺乏严格理论支撑。本文揭示：任何湍流中不稳定的普适因果机制源于**谱偏差**——即**任何**深度学习架构都会偏向仅学习大尺度动力学而完全忽略小尺度特征。我们进一步阐明湍流物理机制与深度学习时间积分器收敛性的缺失如何放大该偏差，导致不稳定的误差传播。最终，以准地转流和ECMWF再分析数据为测试案例，我们弥合深度学习理论与基础数值分析之间的鸿沟，提出此类不稳定性的一种缓解方案。我们开发了针对气候系统的长期稳定数据驱动数字孪生，验证了其短期预报的准确性，以及数百年时间积分中均值与变率均保持精确的长期稳定性能。