Learning Calibrated and Samplable Predictive Distributions of Surface-Ocean $\Delta p\mathrm{CO}_2$ Using Tabular Diffusion Models

The ocean absorbs almost $30$% of anthropogenic $\mathrm{CO}_2$ emissions, but quantifying this sink requires estimating surface-ocean $\mathrm{CO}_2$ from sparse observations in space and time. Existing machine-learning estimates of this field are deterministic, giving a single value per grid point, with uncertainty approximated post hoc via residuals, ensembles, or prescribed distributions. We instead learn a distributional representation of the field: using Treeffuser, a conditional diffusion model with gradient-boosted-tree score functions, we estimate the full predictive distribution of $\Delta p\mathrm{CO}_2$ (the sea–air $p\mathrm{CO}_2$ difference) from $13$ environmental, spatial, and temporal predictors. On a held-out SOCAT test set, Treeffuser attains, among the uncertainty-quantification methods evaluated, the best continuous ranked probability score (8.229 $\micro atm$) and coverage $0.686$ / $0.930$ closest to $68$% / $95$% nominal levels respectively, while retaining point-prediction skill comparable to a deterministic XGBoost baseline (RMSE $18.4$ vs $17.7$ $\micro atm$; $R^2$ $0.77$ vs $0.79$). These results demonstrate that conditional diffusion can provide accurate observation-specific uncertainties while largely preserving point-prediction skill, supporting probabilistic studies of ocean carbon variability. Because of its generative nature, Treeffuser also allows one to sample from the learned distribution of $\Delta p\mathrm{CO}_2$, unlocking new tools for out-of-distribution reconstruction of the ocean carbon field.