An artificial neural network as a troubled-cell indicator

Abstract

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.

Detecting troubled-cells on two-dimensional unstructured grids using a neural network

“…Thus, it is highly desirable to use a troubled-cell indicator free from problem-dependent parameters. This objective was achieved in [34] for one-dimensional conservation laws, by training an feedforward neural network to detect troubled-cells. In Section 4, we extend this approach and propose a similar network for two-dimensional problems.…”

“…While this is a promising attempt at removing the dependence on parameters, it has been demonstrated to perform well only on uniform grids. In [34], a new approach for constructing troubled-cell indicators was proposed using machine learning ideas. In particular, a deep fully-connected feedforward network (or multilayer perceptron) was trained offline to flag discontinuities in the mesh.…”

“…In this work, we extend the machine learning techniques of [34] to train a suitable network to work as a troubled-cell indicator for two-dimensional conservation laws on unstructured triangular grids. To make the method cost effective on two-dimensional grids, the size of the network is drastically reduced as compared to that used in [34]. The results obtained with the network are compared to those obtained with the TVB indicator.…”

Machine Learning design of Volume of Fluid schemes for compressible flows

“…Machine Learning (ML) [19] for the construction of numerical fluxes adapted to Finite Volume (FV) discretizations is becoming a research subject of its own, see recent contributions for the discretization of hyperbolic equations by Hesthaven et al [30,34]. For viscous incompressible flows, like bubble flows where the curvature of the interface controls the dynamics, it seems that ML techniques are established techniques now [37]: we refer to Zaleski et al [28,2] for the reconstruction of the curvature of interfaces and to [16] for an extension to compressible effects.…”

“…This brief tour of ML features and Lagrange+remap features motivates the development of a ML flux function which aims at an accurate transport/remap of a reconstructed interface. Following Hesthaven's approach [30,34], we focus on the numerical construction of a flux function where the inputs contain the local volume fractions and output is the flux. That is the reconstruction of the interface features (in terms of curvature, angle at the corners, .…”