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Time-frequency distributions (TFDs) play a vital role in providing descriptive analysis of non-stationary signals involved in realistic scenarios. It is well known that low time-frequency (TF) resolution and the emergency of cross-terms (CTs) are two main issues, which make it difficult to analyze and interpret practical signals using TFDs. In order to address these issues, we propose the U-Net aided iterative shrinkage-thresholding algorithm (U-ISTA) for reconstructing a near-ideal TFD by exploiting structured sparsity in signal TF domain. Specifically, the signal ambiguity function is firstly compressed, followed by unfolding the ISTA as a recurrent neural network. To consider continuously distributed characteristics of signals, a structured sparsity constraint is incorporated into the unfolded ISTA by regarding the U-Net as an adaptive threshold block, in which structure-aware thresholds are learned from enormous training data to exploit the underlying dependencies among neighboring TF coefficients. The proposed U-ISTA model is trained by both non-overlapped and overlapped synthetic signals including closely and far located non-stationary components. Experimental results demonstrate that the robust U-ISTA achieves superior performance compared with state-of-the-art algorithms, and gains a high TF resolution with CTs greatly eliminated even in low signal-to-noise ratio (SNR) environments.