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Physics-guided machine learning (PGML) has become a prevalent approach in studying scientific systems due to its ability to integrate scientific theories for enhancing machine learning (ML) models. However, most PGML approaches are tailored to isolated and relatively simple tasks, which lim- its their applicability to complex systems involving multiple interacting processes and numerous influencing features. In this paper, we propose a Physics-Guided Foundation Model (PGFM) that combines pre-trained ML models and physics- based models and leverages their complementary strengths to improve the modeling of multiple coupled processes. To effectively conduct pre-training, we construct a simulated en- vironmental system that encompasses a wide range of influ- encing features and various simulated variables generated by physics-based models. The model is pre-trained in this sys- tem to adaptively select important feature interactions guided by multi-task objectives. We then fine-tune the model for each specific task using true observations, while maintaining con- sistency with established physical theories, such as the prin- ciples of mass and energy conservation. We demonstrate the effectiveness of this methodology in modeling water temper- ature and dissolved oxygen dynamics in real-world lakes. The proposed PGFM is also broadly applicable to a range of sci- entific fields where physics-based models are being used. 

Abstract Lake water clarity, phytoplankton biomass, and hypolimnetic oxygen concentration are metrics of water quality that are highly degraded in eutrophic systems. Eutrophication is linked to legacy nutrients stored in catchment soils and in lake sediments. Long lags in water quality improvement under scenarios of nutrient load reduction to lakes indicate an apparent ecosystem memory tied to the interactions between water biogeochemistry and lake sediment nutrients. To investigate how nutrient legacies and ecosystem memory control lake water quality dynamics, we coupled nutrient cycling and lake metabolism in a model to recreate long‐term water quality of a eutrophic lake (Lake Mendota, Wisconsin, USA). We modeled long‐term recovery of water quality under scenarios of nutrient load reduction and found that the rates and patterns of water quality improvement depended on changes in phosphorus (P) and organic carbon storage in the water column and sediments. Through scenarios of water quality improvement, we showed that water quality variables have distinct phases of change determined by the turnover rates of storage pools—an initial and rapid water quality improvement due to water column flushing, followed by a much longer and slower improvement as sediment P pools were slowly reduced. Water clarity, phytoplankton biomass, and hypolimnetic dissolved oxygen differed in their time responses. Water clarity and algal biomass improved within years of nutrient reductions, but hypolimnetic oxygen took decades to improve. Even with reduced catchment loading, recovery of Lake Mendota to a mesotrophic state may require decades due to nutrient legacies and long ecosystem memory. 

Abstract Hybrid Knowledge‐Guided Machine Learning (KGML) models, which are deep learning models that utilize scientific theory and process‐based model simulations, have shown improved performance over their process‐based counterparts for the simulation of water temperature and hydrodynamics. We highlight the modular compositional learning (MCL) methodology as a novel design choice for the development of hybrid KGML models in which the model is decomposed into modular sub‐components that can be process‐based models and/or deep learning models. We develop a hybrid MCL model that integrates a deep learning model into a modularized, process‐based model. To achieve this, we first train individual deep learning models with the output of the process‐based models. In a second step, we fine‐tune one deep learning model with observed field data. In this study, we replaced process‐based calculations of vertical diffusive transport with deep learning. Finally, this fine‐tuned deep learning model is integrated into the process‐based model, creating the hybrid MCL model with improved overall projections for water temperature dynamics compared to the original process‐based model. We further compare the performance of the hybrid MCL model with the process‐based model and two alternative deep learning models and highlight how the hybrid MCL model has the best performance for projecting water temperature, Schmidt stability, buoyancy frequency, and depths of different isotherms. Modular compositional learning can be applied to existing modularized, process‐based model structures to make the projections more robust and improve model performance by letting deep learning estimate uncertain process calculations.