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. Most 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, for each observation, a conditional predictive distribution: 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 lowest continuous ranked probability score ($8.229$ $\micro atm$ and the coverage closest to the nominal 68\% / 95\% levels ($0.686$ / $0.930$), 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 show that conditional diffusion can provide well-calibrated, observation-level uncertainties while largely preserving point-prediction skill. Because Treeffuser is generative, its samples can support probabilistic analyses of $\Delta p\mathrm{CO}_2$; extending this to spatially coherent field reconstruction will require modeling dependence across grid cells and evaluation on spatially held-out data.