Huan Liu

Huan Liu, Hong Wang, Xiangcheng Zheng

The fractional advection-dispersion equation (FADE) has attracted increased attention from researchers as it provides an accurate description for challenging phenomenas with long-range time memory and spatial interactions, such as the anomalous diffusion behavior in the solute transport in porous media. Practically, a full characterization of the model parameters, such as the fluid velocity, dispersion coefficient and the order of the fractional derivative, often implies a huge amount of experiments and measurements and thus are hard to be determined. In this paper, we employ the framework of feedforward deep neural networks (DNNs) to develop an efficient data-driven deep learning algorithm for inferring the aforementioned parameters of the FADE, such as the time-dependent space-fractional advection-dispersion equation (sFADE) and the variable-order fractional mobile/immobile equation (VoFMIE), in which the feedforward DNNs are trained to minimize the mean square error loss function formulated by means of the finite difference approximations of sFADE and VoFMIE, respectively. Several numerical experiments, in which we discover the model parameters by the feedforward DNNs for both the synthetic and field data, are presented to demonstrate the effectiveness and robustness of the proposed data-driven deep learning algorithm.

Jun Chen, Jia Wang, Ruibin Li et al.

Optimal transport has found widespread applications in signal processing and machine learning. Among its many equivalent formulations, optimal transport seeks to reconstruct a random variable/vector with a prescribed distribution at the destination while minimizing the expected distortion relative to a given random variable/vector at the source. However, in practice, certain constraints may render the optimal transport plan infeasible. In this work, we consider three types of constraints: rate constraints, dimension constraints, and channel constraints, motivated by perception-aware lossy compression, generative principal component analysis, and deep joint source-channel coding, respectively. Special attenion is given to the setting termed Gaussian Wasserstein optimal transport, where both the source and reconstruction variables are multivariate Gaussian, and the end-to-end distortion is measured by the mean squared error. We derive explicit results for the minimum achievable mean squared error under the three aforementioned constraints when the covariance matrices of the source and reconstruction variables commute.