ontrolling oscillations in high-order accurate methods through neural networks
While discontinuous Galerkin methods have proven themselves to be powerful computational methods, capable of accurately solving a variety of PDE's, the combination of high-order accuracy and discontinuous solutions remain a significant challenge. Traditional methods such as TVB limiting or artificial viscosity methods have several disadvantages, e.g., a need to specify one or several parameters or the complexity of the methods to avoid overdissipation.
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In this talk we discuss recent developments in which an artificial neural network is used as a troubled cell indicator in limiter based methods or to estimate the nonlinear viscosity in artificial viscosity methods. The neural network is trained independently and is therefore not problem dependent.
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Extensive computational results in one- and two-dimensions shall demonstrate theÌý efficiency of such techniques which, as we shall likewise demonstrate, are often both superior and faster than traditional techniques.
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This work is done in collaboration with D. Ray (EPFL, CH), N. Discacciati (EPFL, CH) and J. Yu (Beihang, PRC).