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To help understand our universe better, researchers and scientists currently run extreme-scale cosmology simulations on leadership supercomputers. However, such simulations can generate large amounts of scientific data, which often result in expensive costs in data associated with data movement and storage. Lossy compression techniques have become attractive because they significantly reduce data size and can maintain high data fidelity for post-analysis. In this paper, we propose to use GPU-based lossy compression for extreme-scale cosmological simulations. Our contributions are threefold: (1) we implement multiple GPU-based lossy compressors to our opensource compression benchmark and analysis framework named Foresight; (2) we use Foresight to comprehensively evaluate the practicality of using GPU-based lossy compression on two real-world extreme-scale cosmology simulations, namely HACC and Nyx, based on a series of assessment metrics; and (3) we develop a general optimization guideline on how to determine the best-fit configurations for different lossy compressors and cosmological simulations. Experiments show that GPU-based lossy compression can provide necessary accuracy on post-analysis for cosmological simulations and high compression ratio of 5~15x on the tested datasets, as well as much higher compression and decompression throughput than CPU-based compressors.