A Deep Smoothness WENO Method with Applications in Option Pricing

Abstract

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.

Efficient WENO Schemes for Nonuniform Grids

“…[23] mentioned by [20]) or finance (see e.g. [14]). Recently, a vast quantity of new WENO variants have been proposed to enhance the accuracy of the traditional WENO scheme proposed in [21,22] as, for instance, Mapped WENO (WENO-M) by Henrick et al [11], WENO-Z [8] or a WENO scheme with progressive order of accuracy close to discontinuities [1].…”