Estimating carbon fluxes over North America using a physics-constrained deep learning model

Quantifying terrestrial ecosystem carbon fluxes, including net ecosystem exchange (NEE) of CO₂, gross primary production (GPP), and ecosystem respiration (RECO), is critical for understanding the global carbon cycle. Although machine learning methods have shown promising performance for this purpose, existing approaches pay less attention to the temporal dynamics of fluxes and often estimate NEE, GPP, and RECO independently without considering their physical relationships. We developed a time-series deep learning model to jointly estimate NEE, GPP, and RECO while accounting for physical relation constraints (i.e., NEE = RECO-GPP). The deep learning model is based on the Transformer architecture and is trained using MODIS variables (nadir BRDF-adjusted reflectance, leaf area index, the fraction of absorbed photosynthetically active radiation) and meteorological variables (air temperature, solar radiation, and soil water content) as predictors, and the three flux measurements obtained from 258 flux tower sites over North America. The model's performance was compared quantitatively and qualitatively against transformer-based, the commonly used extreme gradient boosting (XGBoost), and Random Forest (RF) methods. Spatial cross-validation results indicate that the physics-constrained transformer model outperformed the transformer three-variable model with RMSE reduction by 3.5%, 1.3%, and 1.5%, outperformed transformer single-variable model by 3.5%, 3.1%, and 2.3%, outperformed XGBoost by 3.5%, 3.1%, and 3.0%, and outperformed RF by 3.5%, 2.5%, and 3.0% in estimating NEE, GPP, and RECO, respectively. The physical constraint reduced the mean absolute difference between NEE and (RECO-GPP) in the time series model by >35 times compared to other models without physical constraint. The physics-constrained transformer model was used to upscale annual NEE, GPP, and RECO simultaneously across North America at 500 m resolution, with upscaled NEE estimates nearly equal to RECO – GPP. Results underscore the model's potential to generate spatially and temporally continuous and physically consistent maps of ecosystem carbon fluxes.