Nonlinear dynamical systems exposed to changing forcing values can exhibit catastrophic transitions between distinct states. The phenomenon of critical slowing down can help anticipate such transitions if caused by a bifurcation and if the change in forcing is slow compared with the system’s internal timescale. However, in many real-world situations, these assumptions are not met and transitions can be triggered because the forcing exceeds a critical rate. For instance, the rapid pace of anthropogenic climate change compared with the internal timescales of key Earth system components, like polar ice sheets or the Atlantic Meridional Overturning Circulation, poses significant risk of rate-induced tipping. Moreover, random perturbations may cause some trajectories to cross an unstable boundary whereas others do not—even under the same forcing. Critical-slowing-down-based indicators generally cannot distinguish these cases of noise-induced tipping from no tipping. This severely limits our ability to assess the tipping risks and to predict individual trajectories. To address this, we make the first attempt to develop a deep learning framework predicting the transition probabilities of dynamical systems ahead of rate-induced transitions. Our method issues early warnings, as demonstrated on three prototypical systems for rate-induced tipping subjected to time-varying equilibrium drift and noise perturbations. Exploiting explainable artificial intelligence methods, our framework captures the fingerprints for the early detection of rate-induced tipping, even with long lead times. Our findings demonstrate the predictability of rate-induced and noise-induced tipping, advancing our ability to determine safe operating spaces for a broader class of dynamical systems than possible so far.

暴露于变化的力值的非线性动力学系统可能会在不同状态之间表现出灾难性的转变。如果由分岔引起的，并且强迫的变化与系统的内部时间尺度相比很慢，临界减速现象可以帮助预测这种转变。然而，在许多实际情况下，这些假设并未得到满足，并且可能会触发转换，因为强迫超过临界速率。例如，与地球系统关键组成部分（如极地冰盖或大西洋经向翻转环流）的内部时间尺度相比，人为气候变化的快速速度构成了速率诱发倾翻的重大风险。此外，随机扰动可能会导致一些轨迹越过不稳定的边界，而另一些轨迹则不会——即使在相同的强迫下也是如此。基于临界减速的指标通常无法区分这些噪声引起的倾翻和无倾翻的情况。这严重限制了我们评估小费风险和预测个人轨迹的能力。为了解决这个问题，我们首次尝试开发一个深度学习框架，在速率诱导的转换之前预测动态系统的转换概率。我们的方法发出早期警告，如在受时变平衡漂移和噪声扰动影响的速率诱导倾翻的三个原型系统上所展示的那样。利用可解释的人工智能方法，我们的框架可以捕获指纹，以便及早检测利率引起的小费，即使交货时间很长。 我们的研究结果表明，速率诱导和噪声诱导的倾翻是可预测的，这提高了我们为更广泛的动力系统确定安全操作空间的能力。