Temporal–Spatial Partial Differential Equation Modeling of Land Cover Dynamics via Satellite Image Time Series and Sparse Regression

Land cover dynamics play a critical role in understanding environmental changes, but accurately modeling these dynamics remains a challenge due to the complex interactions between temporal and spatial factors. In this study, we propose a novel temporal–spatial partial differential equation (TS-PDE) modeling method combining sparse regression to uncover the governing equations behind long-term satellite image time series. By integrating temporal and spatial differential terms, the TS-PDE framework captures the intricate interactivity of these factors, overcoming the limitations of traditional pixel-wise prediction methods. Our approach leverages 1×1 convolutional kernels within a convolutional neural network (CNN) solver to approximate derivatives, enabling the discovery of interpretable equations that generalize across temporal–spatial domains. Using MODIS and Planet satellite data, we demonstrate the effectiveness of the TS-PDE method in predicting the value of the normalized difference vegetation index (NDVI) and interpreting the physical significance of the derived equations. The numerical results show that the model achieves good performance, with mean structural similarity index (SSIM) values exceeding 0.82, mean peak signal-to-noise ratio (PSNR) values ranging from 28.5 to 32.8, and mean mean squared error (MSE) values approximating 9×10−4 for low-resolution MODIS images. For high-resolution Planet images, this study emphasizes the efficacy of TS-PDE in terms of PSNR, SSIM, and MSE metrics, with all datasets exhibiting an average SSIM value of over 0.81, an average PSNR maximum of 30.9, and an average MSE of less than 0.0042. The experimental findings demonstrate the capability of TS-PDE in deriving governing equations and providing effective predictions for the regional-scale dynamics of these time series images. The findings of this study provide potential insights into the mathematical modeling of land cover dynamics.