Supplementary Information for Data-driven predictions of the time remaining until critical global warming thresholds are reached

This supplementary information supports an analysis using data-driven methods to predict the time remaining until critical global warming thresholds are reached. The focus is on evaluating predictive performance across scenarios and datasets, assessing uncertainty and sensitivity, and detailing model architecture and validation approaches.

- Predictive framework: Artificial neural networks (ANNs) are trained to predict the time until specific global warming thresholds (e.g., 1.1°C, 1.3°C, 1.5°C, 2.0°C above a pre-industrial baseline of 1850–1899) are reached. - Training data: ANNs are trained using global climate model (GCM) simulations under multiple Shared Socioeconomic Pathway (SSP) forcing scenarios: Low (SSP1-2.6), Intermediate (SSP2-4.5), and High (SSP3-7.0). No historical temperature observations are used in training, validation, or testing. - Inputs and labels: Inputs are maps of annual temperature anomalies; labels include the year a given threshold is reached and, in an alternative formulation, the amount of additional warming until the peak forced global-mean temperature (for SSP1-2.6). - Observational and reanalysis datasets for inference: Predictions are generated using observed/reanalysis maps from Berkeley Earth, GISTEMP, ERA5, and NCEP/NCAR Reanalysis 1 (NCEP1), including for the year 2021 and for annual sequences from 1970–2021. - Scenarios and thresholds evaluated: Predictions are provided for multiple thresholds and SSP scenarios, including additional panels for SSP2-4.5 and SSP1-2.6, and for the 2.0°C threshold where some GCMs in SSP1-2.6 never reach 2.0°C (out-of-sample tests). - Ensemble handling and smoothing: Results are shown for individual realizations and ensemble means of GCMs; a sensitivity variant defines labels using ensemble mean time series smoothed with a Savitzky–Golay filter (window length 15 years, polynomial order 3). - Hyperparameter tuning: Sensitivity analyses include sweeps over ridge regularization parameter, number of hidden layers, and number of hidden nodes. Impacts of hyperparameter choices on 2021 predictions (e.g., with Berkeley Earth) are assessed. - Calibration and uncertainty assessment: Probability integral transform (PIT) histograms are presented for validation and testing data (e.g., best validation seed for 1.5°C in SSP3-7.0, seed=2247). Distributions (PDFs) of predicted threshold years are shown. - Explainable AI (XAI): Three different XAI methods are applied to cases near the 1.5°C threshold (e.g., members within 21–25 years and 1–5 years of the threshold) and across SSPs to interpret model behavior. - Sensitivity to model ensembles: Sensitivity testing examines inclusion/exclusion of high-transient climate response (TCR) GCMs in SSP1-2.6, with “high-TCR” defined as TCR ≥ 2.0°C. - Robustness checks: Range of predictions across random seeds, cross-dataset comparisons of anomalies (including 1951–1980 baseline differences), and time-to-threshold predictions across 1970–2021 are reported.

- ANN predictions using observed/reanalysis 2021 anomaly maps (Berkeley, GISTEMP, ERA5, NCEP1) provide distributions of years when thresholds such as 1.1°C and 1.3°C are reached; historical observations are not used for training/validation/testing. - Under SSP1-2.6, the ANN trained to predict additional warming until the peak forced global mean temperature finds, for 2021 inputs, a predicted additional warming range of approximately 1.1°C to 1.2°C across Berkeley, GISTEMP, ERA5, and NCEP1. The associated prediction standard deviations are approximately 0.23°C to 0.27°C. Excluding NCEP1 narrows the prediction range to about 1.1°C to 1.19°C with standard deviations about 0.23°C to 0.25°C. - Some GCMs in SSP1-2.6 never reach 2.0°C in the 21st century; out-of-sample prediction behavior for the 2.0°C threshold in these cases is evaluated (three such models highlighted), with individual realizations and single-model means shown. - Cross-dataset consistency and differences: Time series and maps reveal differences among datasets (Berkeley, GISTEMP, ERA5, NCEP1) in 2021 anomalies; prediction ranges across datasets and random seeds are documented. - Hyperparameter choices (ridge regularization, network depth/width) influence predictions, but chosen configurations are supported by validation/testing performance and PIT calibration diagnostics. - Sensitivity tests show the influence of including high-TCR (≥2.0°C) GCMs in the SSP1-2.6 training ensemble on predicted outcomes.

The supplementary analyses demonstrate that a data-driven ANN trained on GCM simulations can use spatial maps of annual temperature anomalies to predict the time remaining until critical warming thresholds. Predictions generated from observed and reanalysis maps yield consistent distributions across multiple datasets, supporting robustness. Evaluation across SSPs, thresholds, and ensemble realizations shows the method’s applicability under a range of forced climate trajectories. Calibration diagnostics (PIT) and hyperparameter sweeps indicate the model can provide well-calibrated probabilistic outputs with stable performance under selected configurations. Sensitivity analyses, including the handling of high-TCR models and label smoothing via ensemble means, clarify how training set composition and label definition affect predictions. Out-of-sample testing for thresholds not reached in some scenarios (e.g., 2.0°C under SSP1-2.6) further probes generalization behavior.

This supplement details model configuration, validation, and robustness checks for ANN-based predictions of time to global warming thresholds. Key contributions include cross-dataset application to observations/reanalyses, uncertainty quantification via PDFs and PIT analysis, and sensitivity testing to hyperparameters, ensemble composition (including high-TCR models), and label definitions. Results indicate broadly consistent and calibrated predictions across datasets and scenarios. Future work could extend to additional thresholds, broaden GCM ensembles, and further refine interpretability analyses; however, such directions are not elaborated in the text.

- The predictive models are trained solely on GCM simulations; no historical observations are used in training, validation, or testing, which may limit transferability if model–data discrepancies exist. - Results depend on the choice of forcing scenario (SSP1-2.6, SSP2-4.5, SSP3-7.0) and the composition of the GCM training ensemble, including sensitivity to inclusion of high-TCR models (≥2.0°C). - Differences among observational and reanalysis datasets (e.g., Berkeley, GISTEMP, ERA5, NCEP1) introduce variation in predicted timelines. - Some tabulated sensitivity results are scenario- and model-specific, and exact quantitative outcomes for certain models/metrics are not fully detailed in the provided text fragments.