Estimation of AMOC transition probabilities using a machine learning based rare-event algorithm

TL;DR

The paper tackles the challenge of estimating AMOC tipping probabilities within a finite time window by coupling the rare-event sampler TAMS with Next-Generation Reservoir Computing to learn the committor function on the fly. Using a conceptual five-box AMOC model, the authors demonstrate that TAMS-R can reproduce transition probabilities, MFPTs, and transition paths for fast F-transitions and extend the approach to slow S-transitions, achieving consistency with physics-informed scores and analytic interpretability of the learned committor. A key contribution is showing that the RC not only matches probabilistic estimates but also yields a transparent, analytical-style understanding of the committor structure, revealing salt-advection feedback as a central mechanism. The work suggests a practical path toward applying rare-event techniques to higher-dimensional climate models, while acknowledging scalability challenges and proposing future avenues such as CNN-based committor models and dimensionality reduction for complex GCMs.

Abstract

The Atlantic Meridional Overturning Circulation (AMOC) is an important component of the global climate, known to be a tipping element, as it could collapse under global warming. The main objective of this study is to compute the probability that the AMOC collapses within a specified time window, using a rare-event algorithm called Trajectory-Adaptive Multilevel Splitting (TAMS). However, the efficiency and accuracy of TAMS depend on the choice of the score function. Although the definition of the optimal score function, called ``committor function" is known, it is impossible in general to compute it a priori. Here, we combine TAMS with a Next-Generation Reservoir Computing technique that estimates the committor function from the data generated by the rare-event algorithm. We test this technique in a stochastic box model of the AMOC for which two types of transition exist, the so-called F(ast)-transitions and S(low)-transitions. Results for the F-transtions compare favorably with those in the literature where a physically-informed score function was used. We show that coupling a rare-event algorithm with machine learning allows for a correct estimation of transition probabilities, transition times, and even transition paths for a wide range of model parameters. We then extend these results to the more difficult problem of S-transitions in the same model. In both cases of F-transitions and S-transitions, we also show how the Next-Generation Reservoir Computing technique can be interpreted to retrieve an analytical estimate of the committor function.

Figures (9)

• Figure 1: Summarizing picture of the AMOC model, in its version from Castellana. Blue arrows represent the freshwater forcings, red arrows represent the volume transports and cyan arrows stand for wind-driven transports. Solid arrows are always present whatever the AMOC regime, dashed arrows correspond to the present-day AMOC regime and dotted arrows represent the fully collapsed AMOC.
• Figure 2: (a) Estimated transition probabilities using TAMS-$R$ depending on the model parameters $(\overline{E_a}, f_\sigma)$. This reproduces the main result of Castellana, who used TAMS-$S$. (b) Consistency score $\mathcal{C}$ for every value of the couple of parameters $(\overline{E_a}, f_\sigma)$. The colorbar in this figure is centered on $1$, which means that in blue grid cells, $\alpha_S^F$ and $\alpha_R^F$ are within each other's $95\%$ confidence interval. When this is not the case, the grid cell appears in red. Gray cells mean that $\alpha_S^F$ or $\alpha_R^F$ was below the $10^{-9}$ cut-off threshold.
• Figure 3: Each panel shows, for a fixed $f_\sigma$, the relative difference of the variance of $30$ runs of TAMS to the corresponding ideal variance $(\sigma^2)_{\mathrm{id}_{\{R,S\}}}^F$. It is plotted against the corresponding mean probability estimate $\alpha_{\{R,S\}}^F$. The blue line corresponds to $V_{\mathrm{diff},S}$ against $\alpha_S^F$. Each thin black line corresponds to $V_{\mathrm{diff},R_{i,i\in[1,30]}}$ against $\alpha_{R_{i, i\in[1,30]}}^F$. The thicker black line represents the average of the thinner black lines.
• Figure 4: a): $1,000$ trajectories have been sampled from $30$ runs of TAMS-$S$. The transition times have been averaged over this ensemble for different values of $(\overline{E_a},f_\sigma)$. This panel shows the mean first-passage times $\mathrm{MFPT}_S$ thus obtained, expressed in model years. b): The MFPT has also been computed for the TAMS runs with the RC, and this panel shows the relative difference $|\mathrm{MFPT}_S-\mathrm{MFPT}_R|/\mathrm{MFPT}_S$, plotted against the mean transition probability $\alpha_S^F$.
• Figure 5: Ensemble of $1000$ trajectories obtained with TAMS-$S$ (in blue) and TAMS-$R$ (in red), for $(\overline{E_a},f_\sigma)=(0.288\mathrm{\ Sv},0.05)$ (corresponding to $\alpha_S^F=4\times10^{-4}$). We present here the average of the trajectories that reached $B$ between model years $90$ and $100$. The shaded areas represent the difference between the $5^\mathrm{th}$ and $95^\mathrm{th}$ percentiles in each case.