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Abstract. Photosynthesis plays an important role in carbon,nitrogen, and water cycles. Ecosystem models for photosynthesis arecharacterized by many parameters that are obtained from limited in situmeasurements and applied to the same plant types. Previous site-by-sitecalibration approaches could not leverage big data and faced issues likeoverfitting or parameter non-uniqueness. Here we developed an end-to-endprogrammatically differentiable (meaning gradients of outputs to variablesused in the model can be obtained efficiently and accurately) version of thephotosynthesis process representation within the Functionally AssembledTerrestrial Ecosystem Simulator (FATES) model. As a genre ofphysics-informed machine learning (ML), differentiable models couplephysics-based formulations to neural networks (NNs) that learn parameterizations(and potentially processes) from observations, here photosynthesis rates. Wefirst demonstrated that the framework was able to correctly recover multiple assumedparameter values concurrently using synthetic training data. Then, using areal-world dataset consisting of many different plant functional types (PFTs), welearned parameters that performed substantially better and greatly reducedbiases compared to literature values. Further, the framework allowed us togain insights at a large scale. Our results showed that the carboxylationrate at 25 ∘C (Vc,max25) was more impactful than a factorrepresenting water limitation, although tuning both was helpful inaddressing biases with the default values. This framework could potentiallyenable substantial improvement in our capability to learn parameters andreduce biases for ecosystem modeling at large scales. 

Process-based modelling offers interpretability and physical consistency in many domains of geosciences but struggles to leverage large datasets efficiently. Machine-learning methods, especially deep networks, have strong predictive skills yet are unable to answer specific scientific questions. In this Perspective, we explore differentiable modelling as a pathway to dissolve the perceived barrier between process-based modelling and machine learning in the geosciences and demonstrate its potential with examples from hydrological modelling. ‘Differentiable’ refers to accurately and efficiently calculating gradients with respect to model variables or parameters, enabling the discovery of high-dimensional unknown relationships. Differentiable modelling involves connecting (flexible amounts of) prior physical knowledge to neural networks, pushing the boundary of physics-informed machine learning. It offers better interpretability, generalizability, and extrapolation capabilities than purely data-driven machine learning, achieving a similar level of accuracy while requiring less training data. Additionally, the performance and efficiency of differentiable models scale well with increasing data volumes. Under data-scarce scenarios, differentiable models have outperformed machine-learning models in producing short-term dynamics and decadal-scale trends owing to the imposed physical constraints. Differentiable modelling approaches are primed to enable geoscientists to ask questions, test hypotheses, and discover unrecognized physical relationships. Future work should address computational challenges, reduce uncertainty, and verify the physical significance of outputs.