Learning Wave Propagation with Attention-Based Convolutional Recurrent Autoencoder Net

In this paper, we present an end-to-end attention-based convolutional recurrent autoencoder network (AB-CRAN) for data-driven modeling of wave propagation phenomena. To construct the low-dimensional learning model, we employ a denoising-based convolutional autoencoder from the full-order snapshots of wave propagation generated by solving hyperbolic partial differential equations. The proposed deep neural network architecture relies on the attention-based recurrent neural network with long short-term memory cells. We assess the proposed AB-CRAN framework against the recurrent neural network for the low-dimensional learning of wave propagation. To demonstrate the effectiveness of the AB-CRAN model, we consider three benchmark problems namely one-dimensional linear convection, nonlinear viscous Burgers, and a two-dimensional Saint-Venant shallow water system. Using the time-series datasets from the benchmark problems, our novel AB-CRAN architecture accurately captures the wave amplitude and preserves the wave characteristics of the solution for long time horizons. The attention-based sequence-to-sequence network increases the time-horizon of prediction by five times compared to the standard recurrent neural network with long short-term memory cells. Denoising autoencoder further reduces the mean squared error of prediction and improves the generalization capability in the parameter space.Comment: 23 page

Hybrid PDE solver for data-driven problems and modern branching

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European Journal of Applied Mathematics (EJAM

Retinal Fundus Image Enhancement Using the Normalized Convolution and Noise Removing

Retinal fundus image plays an important role in the diagnosis of retinal related diseases. The detailed information of the retinal fundus image such as small vessels, microaneurysms, and exudates may be in low contrast, and retinal image enhancement usually gives help to analyze diseases related to retinal fundus image. Current image enhancement methods may lead to artificial boundaries, abrupt changes in color levels, and the loss of image detail. In order to avoid these side effects, a new retinal fundus image enhancement method is proposed. First, the original retinal fundus image was processed by the normalized convolution algorithm with a domain transform to obtain an image with the basic information of the background. Then, the image with the basic information of the background was fused with the original retinal fundus image to obtain an enhanced fundus image. Lastly, the fused image was denoised by a two-stage denoising method including the fourth order PDEs and the relaxed median filter. The retinal image databases, including the DRIVE database, the STARE database, and the DIARETDB1 database, were used to evaluate image enhancement effects. The results show that the method can enhance the retinal fundus image prominently. And, different from some other fundus image enhancement methods, the proposed method can directly enhance color images

A PDE-based Mathematical Method in Image Processing: Digital-Discrete Method for Perona-Malik Equation

In this study, we propose a new and effective algorithm for image processing. The method based on the combination of digital topology, partial differential equations and finite difference scheme is called the digital-discrete method. We try to solve the Perona-Malik equation using the digital-discrete method. We use the MATLAB package program when analyzing images. The analyzes we make on the images show how the algorithm is useful, effective and open to development