Fractal-Informed Neural Ordinary Differential Equations for Continuous Forecasting of Microalgae Dynamics A Full Experimental Implementation Using Multitemporal Remote Sensing Data

Abstract

Coastal ecosystems in the Southern Mediterranean face intensifying ecological pressures from eutrophication-driven nutrient loading, producing recurrent microalgal blooms with cascading consequences for marine biodiversity, fisheries, and public health. Reliable operational forecasting remains challenging: traditional kinetic models assume spatial homogeneity incompatible with real bloom morphology, while discrete deep-learning architectures introduce temporal smoothing that attenuates sharp bloom peaks. This study presents FI-NODE (Fractal-Informed Neural Ordinary Differential Equations), a hybrid Physics-Informed Machine Learning (PIML) framework that embeds a dynamically evolving fractal dimension Df(t), extracted weekly from Sentinel-3 OLCI Level-3 chlorophyll-a imagery, directly into the differential operator of a continuous-time Neural ODE. To guard against target-derived feature leakage, all reported forecasts use only strictly lagged Df(t) values available at the forecast issue time, and the forecasting protocol (input window, horizon, and lag structure) is stated explicitly. Evaluated on observational data (2018–2023) across five hydrologically distinct regions—including an independent out-of-distribution validation zone in the Black Sea—FI-NODE outperforms state-of-the-art deep time-series forecasting architectures (TimeXer, iTransformer, PatchTST, TimeMixer) and static Physics-Informed Neural Networks (PINNs), with all pairwise comparisons evaluated using autocorrelation-aware, block-bootstrapped statistical tests. The framework achieves a mean R² = 0.891 ± 0.017 and RMSE = 0.89 ± 0.09 mg Chl/m³ on the 2023 held-out test set, a 9.4% RMSE reduction relative to the strongest baseline (TimeMixer) on the primary test region. Global Sobol sensitivity analysis reveals that Df contributes 26% to first-order output variance, ranking second only to sea surface temperature. Probabilistic calibration, evaluated using a regression-appropriate Expected Calibration Error together with prediction-interval coverage and sharpness, yields an ECE of 0.023, supporting the framework's potential utility for risk-based coastal management pending further operational validation. Cross-correlation analysis, accounting for the reduced effective degrees of freedom of temporally autocorrelated weekly series, indicates that Df elevation precedes chlorophyll-a concentration peaks by an estimated 4.2 ± 0.8 days (p < 0.001); because this estimate is derived from interpolated sub-weekly values built on 10-day composite imagery, we treat it as an indicative structural lead time rather than a precisely resolved operational figure, and present it as a candidate early-warning signal for bloom management warranting confirmation with higher-frequency observations.