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While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods as a local estimate of solution regularity generally is required. In this work we demonstrate how to train and use a local artificial neural network to estimate the local solution regularity and demonstrate the efficiency of nonlinear artificial viscosity methods based on this in the context of Fourier spectral methods. We compare with entropy viscosity techniques and illustrate the promise of the neural network based estimators when solving one- and two-dimensional conservation laws, including the Euler equations.