Deep learning for solving partial differential equations (PDEs) has been a major research hotspot. Various neural network frameworks have been proposed to solve nonlinear PDEs. However, most deep learning-based methods need labeled data, while traditional numerical solutions do not need any labeled data. Aiming at deep learning-based methods behaving as traditional numerical solutions do, this paper proposed an approximation-correction model to solve unsteady compressible seepage equations with sinks without using any labeled data. The model contains two neural networks, one for approximating the asymptotic solution, which is mathematically correct when time tends to 0 and infinity, and the other for correcting the error of the approximation, where the final solution is physically correct by constructing the loss function based on the boundary conditions, PDE, and mass conservation. Numerical experiments show that the proposed method can solve seepage equations with high accuracy without using any labeled data, as conventional numerical solutions do. This is a significant breakthrough for deep learning-based methods to solve PDE.
Indoor localization services are emerging as an important application of the Internet of Things, which drives the development of related technologies in indoor scenarios. In recent years, various localization algorithms for different indoor scenarios have been proposed. The indoor localization algorithm based on fingerprints has attracted much attention due to its good performance without extra hardware devices. However, the occurrence of fingerprint mismatching caused by the complexity and variability of indoor scenarios is unneglectable, which degrades localization accuracy. In this article, by combining weighted kernel norm and L2,1-norm, a joint-norm robust principal component analysis (JRPCA in brief) assisted indoor localization algorithm is proposed, which can improve the localization accuracy through aggregating the reference points (RPs) and conducting robust feature extraction based on clustering. More specifically, a one-way hierarchical clustering termination method is proposed to obtain reasonable RP clusters adaptively according to the preset RPs. A two-phase fingerprint matching algorithm of JRPCA based on clustering is proposed to further increase the difference between similar RPs, thus facilitating rapid identification and reinforcing localization accuracy. To validate the proposed algorithm, extensive experiments are conducted in real indoor scenarios. The experimental results confirm that the proposed cluster-based JRPCA algorithm outperforms other existing algorithms in terms of robustness and accuracy.
Hyperspectral images (HSIs) are usually corrupted by various types of mixed noises, which degrades the qualities of acquired images and limits the subsequent applications. In this paper, we propose a novel denoised method based on a hybrid spatial-spectral total variation (SSTV) regularization, which we refer to as l0-l1−2SSTV. Specifically, l0-l1−2SSTV can be treated as an integrated regularization embedding spatial-spectral l0 gradient model into l1−2SSTV. l1−2SSTV regularization exploits the sparse structures in both spatial and spectral domains. Hence, the correlations within HSIs are fully considered. Due to the good performance of l1−2-norm in image restoration, l1−2SSTV gives a tighter approximation for the gradient domains of HSIs. It can effectively avoid artifacts and oversmoothing caused by the limitation of the SSTV regularization based on l1-norm (l1SSTV). Meanwhile, the l0 gradient regularization controls the number of nonzero gradients to promote the local piecewise smoothness, making denoised images preserve clear edges. With the effective combination of l1−2SSTV and l0 gradient regularization, l0-l1−2SSTV produces high-quality restoration results in the denoising process: better detail preservation and sharper edges. The augmented Lagrangian method (ALM) and the difference of convex algorithm (DCA) are exploited to optimize the proposed model. The results for simulated and real experiments demonstrate the effectiveness and superiority of the proposed method compared with state-of-the-art methods.
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