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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 Lévy 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. 

Recycling underutilized resources from food waste (FW) to agriculture through hydrothermal carbonization (HTC) has been proposed to promote a circular economy (CE) in food-energy-water (FEW) nexus. However, most HTC studies on FW were conducted at laboratory scale, and little is known on the efficacy and feasibility of field application of HTC products from FW, i.e. the aqueous phrase (AP) and solid hydrochar (HC), to support agriculture production. An integrated pilot-scale HTC system was established to investigate practical HTC reaction conditions treating FW. A peak temperature of 180 ◦C at a residence time of 60 min with 3 times AP recirculation were recommended as optimal HTC conditions to achieve efficient recovery of nutrients, and desirable AP and HC properties for agriculture application. Dilution of the raw AP and composting of the fresh HC are necessary as post-treatments to eliminate potential phytotoxicity. Applying properly diluted AP and the composted HC significantly improved plant growth and nutrient availability in both greenhouse and field trials, which were comparable to commercial chemical fertilizer and soil amendment. The HTC of FW followed with agricultural application of the products yielded net negative carbon emission of 􀀀 0.28 t CO2e t􀀀 1, which was much lower than the other alternatives of FW treatments. Economic profit could be potentially achieved by valorization of the AP as liquid fertilizer and HC as soil amendment. Our study provides solid evidences demonstrating the technical and economic feasibility of recycling FW to agriculture through HTC as a promising CE strategy to sustain the FEW nexus.