The inviscid Euler equations are a system of PDEs modeling the flow of fluids when the effects of viscosity are ignored.

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Discrete solutions solved on a staggered grid produce spurious, Gibbs-like, oscillations near discontinuities (shocks, for example).

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A classic approach, dating back to VonNeumann & Richtmyer, is to augment the discrete equations with artificial viscosity (AV) to act as a damping mechanism in computational cells under compression.  Reducing oscillations comes at the cost of smearing the shock profile over a finite number of cells.

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Our goal was to train an artificial neural network (ANN) to reduce oscillations and maintain a sharp shock profile.  To train such a network, we embedded a pre-trained ANN model in a numerical scheme and updated ANN parameters by back-propagating through the entire code using the differentiable programming capabilities of the Julia language.

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We found that training over early time steps resulted in a neural AV function that could reduce oscillations in (relative to the training) long-term simulations.

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You can read more about this work here.