This supplementary material expands upon the main research paper, which investigates the prediction of remaining time until critical global warming thresholds are crossed. The central research question is: Given current climate trends and various emission scenarios, when are we likely to surpass key temperature thresholds? The context is the urgent need to understand the timeframe for reaching 1.5°C and 2.0°C of global warming above pre-industrial levels, thresholds identified in the Paris Agreement as crucial to avoiding severe climate change impacts. The importance lies in informing climate policy and adaptation strategies by providing more precise data-driven estimates of the time remaining before these critical thresholds are breached. The study uses a novel data-driven approach, leveraging an artificial neural network (ANN) trained on climate model projections to predict the time to reaching different warming thresholds based on current observed temperatures. This approach allows for consideration of both internal climate variability and the uncertainty inherent in future emissions scenarios.

The supplementary material does not explicitly contain a separate literature review section. However, the main paper likely reviewed existing literature on climate model projections, global warming thresholds, and time-to-exceedance predictions. It implicitly builds on previous research that has explored the uncertainties surrounding future warming projections and the methodologies used to estimate the probability of surpassing certain temperature levels.

The core methodology involves the use of an artificial neural network (ANN) trained on global climate model (GCM) simulations under three Shared Socioeconomic Pathways (SSPs): SSP1-2.6 (low emissions), SSP2-4.5 (intermediate emissions), and SSP3-7.0 (high emissions). The ANN was trained to predict the time until specific global warming thresholds (1.1°C, 1.5°C, and 2.0°C) are reached. The input data for the ANN included temperature anomaly maps generated from the GCMs. Importantly, the ANN was not trained on historical temperature observations; instead, it used only the GCM projections. The supplementary materials demonstrate the robustness of the results by using multiple observational datasets (Berkeley Earth, GISTEMP) and reanalyses (ERA5, NCEP1) to generate independent predictions. Sensitivity analyses were performed to evaluate the impact of different GCMs, model parameters, and datasets on the accuracy and uncertainty of the predictions. Several explainable AI (XAI) methods were applied to investigate the contribution of different factors to the model's predictions. Hyperparameter tuning was conducted to optimize the ANN's performance. Specific details on the ANN architecture (number of hidden layers, nodes), activation functions, and training algorithms are likely provided in the main paper or additional supplementary files.

The supplementary figures and tables provide detailed visualizations and results of the ANN's predictions for the time remaining until various global warming thresholds under different climate forcing scenarios. The key findings are likely to include variations in the predicted time to exceedance based on the emission scenario, the influence of using different datasets (observational and reanalysis) and the uncertainty associated with the model's predictions. Figures S1-S8 and S15 illustrate predicted time to different warming thresholds under various scenarios and datasets. Figures S9 and S10 explore predictions from specific climate models. Figures S11-S14 analyze XAI methods applied to the model. Figures S16-S20 explore the impact of smoothing, hyperparameter tuning, and probabilistic measures on the predictions. Supplemental Table 1 lists the GCMs used for training, while Supplemental Table 2 presents results of sensitivity testing related to the inclusion of high-transient climate response (TCR) GCMs in predictions.

The supplementary materials support the main paper's findings by providing additional evidence for the robustness and reliability of the data-driven approach. The sensitivity analysis clarifies the impact of data choices and model parameters. The use of multiple observational datasets and the exploration of different ANN architectures help to assess and quantify uncertainty in the predictions. The consistent patterns observed across different datasets strengthen the credibility of the predicted timeframes for reaching critical warming thresholds. Further research could focus on incorporating additional climate variables, refining the ANN architecture, or exploring alternative machine-learning techniques to further improve the accuracy and precision of these critical predictions.

The supplementary information provides comprehensive visualizations and sensitivity analyses that enhance the main paper's conclusions. The results consistently demonstrate that exceeding critical global warming thresholds is highly probable within the coming decades under various emission scenarios. The detailed explorations of uncertainty and sensitivity provide valuable context for interpreting and applying these predictions to inform effective climate policy and mitigation strategies. Future research could explore more sophisticated machine learning methods, incorporate a wider range of climate indicators, and refine the quantification of uncertainty.

The limitations of the study likely include the dependence on the accuracy and completeness of the GCM data used for training. The uncertainties associated with future emissions scenarios and the inherent limitations of the ANN model itself also introduce uncertainty into the predictions. The reliance on specific datasets and reanalyses also means the results are not entirely independent of the data selected. These limitations are likely addressed in detail in the discussion section of the main paper, with the supplementary material contributing to a better understanding of their impact.