A neural network enhanced weighted essentially non-oscillatory method for nonlinear degenerate parabolic equations

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.