Controlling oscillations in high-order schemes using neural networks

High-order numerical solvers for conservation laws suffer from Gibbs phenomenon close to discontinuities, leading to spurious oscillations and a detrimental effect on the solution accuracy. A possible strategy to reduce their amplitude aims to add a suitable amount of artificial viscosity. Several models are available in the literature, which rely on the identification of a shock sensor, adding dissipation when the solution regularity is lost. The dependence on problem-specific parameters limits their performances. To solve this issue, in this thesis we propose a new technique based on artificial neural networks. In particular, we focus on the construction of multilayer perceptrons. Emphasis is given to the training phase, which is carried out using a robust dataset created using the classical models with optimal parameters. The online evaluation is then integrated in the numerical solver of the partial differential equation. Even though most of the effort is put in one-dimensional problems, an extension to a two-dimensional scenario is provided. Several numerical results are presented to demonstrate the capabilities of the network-based technique. Initially, we focus on one-dimensional scalar problems, where Burgers equation represents our first benchmark test. Then, we move to less simple cases, characterized by non-convex flux functions or involving multiple equations (compressible Euler equations). The same strategy is followed for multi-dimensional problems. In most of the cases, the proposed model is able to guarantee high accuracy in presence of smooth solutions and to capture discontinuities (shock and contact waves). In general, the results are comparable to (or better than) the classical models with properly tuned parameters. A final performance analysis is carried out.

Controlling oscillations in high-order Discontinuous Galerkin schemes using artificial viscosity tuned by neural networks

Niccolo' Discacciati, Jan Sickmann Hesthaven, Deep Ray

ACADEMIC PRESS INC ELSEVIER SCIENCE2020

MATHICSE Technical Report: Controlling oscillations in high-order Discontinuous Galerkin schemes using artificial\ viscosity tuned by neural networks

Niccolo' Discacciati, Jan Sickmann Hesthaven, Deep Ray

High-order numerical solvers for conservation laws suer from Gibbs phenomenon close to discontinuities, leading to spurious oscillations and a detrimental effect on the solution accuracy. A possible strategy to reduce it comprises adding a suitable amount of artificial dissipation. Although several viscosity models have been proposed in the literature, their dependence on problem-dependent parameters often limits their performances. Motivated by the objective to construct a universal artificial viscosity method, we propose a new technique based on neural networks, integrated into a Runge-Kutta Discontinuous Galerkin solver. Numerical results are presented to demonstrate the performance of this network-based technique. We show that it is able both to guarantee optimal accuracy for smooth problems, and to accurately detect discontinuities, where dissipation has to be injected. A comparison with some classical models is carried out, showing the superior performance of the network-based model in capturing both complex and fine structures.

MATHICSE2019

An artificial neural network as a troubled-cell indicator

Jan Sickmann Hesthaven, Deep Ray

High-resolution schemes for conservation laws need to suitably limit the numerical solution near discontinuities, in order to avoid Gibbs oscillations. The solution quality and the computational cost of such schemes strongly depend on their ability to correctly identify troubled-cells, namely, cells where the solution loses regularity. Motivated by the objective to construct a universal troubled-cell indicator that can be used for general conservation laws, we propose a new approach to detect discontinuities using artificial neural networks (ANNs). In particular, we construct a multilayer perceptron (MLP), which is trained offline using a supervised learning strategy, and thereafter used as a black-box to identify troubled-cells. The proposed MLP indicator can accurately identify smooth extrema and is independent of problem-dependent parameters, which gives it an advantage over traditional limiter-based indicators. Several numerical results are presented to demonstrate the robustness of the MLP indicator in the framework of Runge-Kutta discontinuous Galerkin schemes, and its performance is compared with the minmod limiter and the minmod-based TVB limiter.

2018