Mathematical Modeling of Nutrient Dynamics in Coastal Waters Using Convolutional Neural Networks: A Deep Learning Approach to Eutrophication Forecasting

Abstract

Coastal eutrophication driven by excessive nutrient loading represents one of the most pressing environmental challenges facing marine ecosystems worldwide, with harmful algal bloom frequency increasing nearly five-fold over the past century. Traditional process-based models for simulating nutrient dynamics, while physically rigorous, face fundamental limitations in computational efficiency and predictive accuracy when applied to optically complex coastal waters where key nutrients such as dissolved inorganic nitrogen lack direct spectral signatures. This paper presents a comprehensive framework for the mathematical modeling of nutrient dynamics using Convolutional Neural Networks (CNNs), demonstrating how deep learning architectures can overcome the inherent limitations of conventional approaches through nonlinear feature extraction from remote sensing reflectance data. We synthesize recent advances in CNN-based water quality retrieval, examining 1D-CNN architectures optimized through hybrid metaheuristic algorithms achieving coefficient of determination values exceeding 0.81 for DIN inversion, CNN-LSTM architectures for short-term chlorophyll-a forecasting with 73% relative accuracy, and graph-neural-network-based zoning approaches that reduce computational time from hours to minutes while maintaining Nash-Sutcliffe efficiency above 0.80. The theoretical foundations linking CNN architectures to the mathematical structure of radiative transfer equations are established, demonstrating that the convolution operation naturally approximates the integral operators governing light-nutrient interactions in turbid media. Practical implementation considerations including data preprocessing, hyperparameter optimization strategies, and spatiotemporal prediction frameworks are systematically addressed. Case studies from the Yellow Sea, Zhanjiang Bay, and the Sundarbans demonstrate the transferability and operational viability of these approaches. The findings establish CNNs as a transformative mathematical tool for coastal nutrient forecasting, enabling data-driven management strategies that address the spatial and temporal heterogeneity inherent in coastal ecosystems.