In complex physical systems, conventional differential equations fall short in capturing non-local and memory effects. Fractional differential equations (FDEs) effectively model long-range interactions with fewer parameters. However, deriving FDEs from physical principles remains a significant challenge. This study introduces a stepwise data-driven framework to discover explicit expressions of FDEs directly from data. The proposed framework combines deep neural networks for data reconstruction and automatic differentiation with Gauss-Jacobi quadrature for fractional derivative approximation, effectively handling singularities while achieving fast, high-precision computations across large temporal/spatial scales. To optimize both linear coefficients and the nonlinear fractional orders, we employ an alternating optimization approach that combines sparse regression with global optimization techniques. We validate the framework on various datasets, including synthetic anomalous diffusion data, experimental data on the creep behavior of frozen soils, and single-particle trajectories modeled by L & eacute;vy motion. Results demonstrate the framework's robustness in identifying FDE structures across diverse noise levels and its ability to capture integer-order dynamics, offering a flexible approach for modeling memory effects in complex systems.

Climate change is causing glaciers to retreat across much of the Himalaya, leading to a rapid shift of the vegetation cover to higher altitudes. However, the rate of vegetation shift with respect to glacier retreat, climate change, and topographic parameters is not empirically quantified. Using remote sensing measurements, we estimate (a) the rate of glacier-ice mass loss, (b) the upward vegetation line shift rate, (c) regional greening trends, and (d) a relationship between the factors influencing the greenness of the landscape and vegetation change in the Himalaya. We find that the glacier mass loss rate is 10.9 +/- 1.2 Gt/yr and the mean vegetation line shifts upward in altitude by 7-28 +/- 1.5 m/yr. Considering the land use/land cover change pattern, the grassland area is found to be expanding the most, particularly in the de-glaciated regions. The vegetation change is found to be controlled by soil moisture and slope of the area.