No abstract

In this paper, a new modification of the weighted essentially non-oscillatory (WENO) method for solving nonlinear degenerate parabolic equations is developed using deep learning techniques. To this end, the smoothing indicators of an existing WENO algorithm, which are responsible for measuring the discontinuity of a numerical solution, are modified. This is done in such a way that the consistency and convergence of our new WENO-DS (deep smoothness) method is preserved and can be theoretically proved. A convolutional neural network (CNN) is used and a novel and effective training procedure is presented. Furthermore, it is shown that the WENO-DS method can be easily applied to additional dimensions without the need to retrain the CNN. Our results are presented using benchmark examples of nonlinear degenerate parabolic equations, such as the equation of a porous medium with the Barenblatt solution, the Buckley–Leverett equation, and their extensions in two-dimensional space. It is shown that in our experiments, the new method outperforms the standard WENO method, reliably handles sharp interfaces, and provides good resolution of discontinuities.

In this paper, we introduce an improved version of the fifth-order weighted essentially nonoscillatory (WENO) shock-capturing scheme by incorporating deep learning techniques. The established WENO algorithm is improved by training a compact neural network to adjust the smoothness indicators within the WENO scheme. This modification enhances the accuracy of the numerical results, particularly near abrupt shocks. Unlike previous deep learning-based methods, no additional post-processing steps are necessary for maintaining consistency. We demonstrate the superiority of our new approach using several examples from the literature for the two-dimensional Euler equations of gas dynamics. Through intensive study of these test problems, which involve various shocks and rarefaction waves, the new technique is shown to outperform traditional fifth-order WENO schemes, especially in cases where the numerical solutions exhibit excessive diffusion or overshoot around shocks.

In this paper we enhance the well-known fifth order WENO shock-capturing scheme by using deep learning techniques. This fine-tuning of an existing algorithm is implemented by training a rather small neural network to modify the smoothness indicators of the WENO scheme in order to improve the numerical results especially at discontinuities. In our approach no further post-processing is needed to ensure the consistency of the method, which simplifies the method and increases the effect of the neural network. Moreover, the convergence of the resulting scheme can be theoretically proven.We demonstrate our findings with the inviscid Burgers' equation, the Buckley-Leverett equation and the 1-D Euler equations of gas dynamics. Hereby we investigate the classical Sod problem and the Lax problem and show that our novel method outperforms the classical fifth order WENO schemes in simulations where the numerical solution is too diffusive or tends to overshoot at shocks.