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Enhancement of shock-capturing methods via machine learning

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Abstract

In recent years, machine learning has been used to create data-driven solutions to problems for which an algorithmic solution is intractable, as well as fine-tuning existing algorithms. This research applies machine learning to the development of an improved finite-volume method for simulating PDEs with discontinuous solutions. Shock-capturing methods make use of nonlinear switching functions that are not guaranteed to be optimal. Because data can be used to learn nonlinear relationships, we train a neural network to improve the results of a fifth-order WENO method. We post-process the outputs of the neural network to guarantee that the method is consistent. The training data consist of the exact mapping between cell averages and interpolated values for a set of integrable functions that represent waveforms we would expect to see while simulating a PDE. We demonstrate our method on linear advection of a discontinuous function, the inviscid Burgers’ equation, and the 1-D Euler equations. For the latter, we examine the Shu–Osher model problem for turbulence–shock wave interactions. We find that our method outperforms WENO in simulations where the numerical solution becomes overly diffused due to numerical viscosity.

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Notes

  1. Note that the error oscillates between two different values because in the exact solution the discontinuity switches between being on the edge of a cell and 1/3 of a cell width away from either the left or right of a cell edge since the CFL number is 2/3. To get a smooth curve, we apply a filter to the error and plot \(E(i) = \frac{e(i)+e(i-1)+e(i-2)}{3}\).

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Authors and Affiliations

  1. Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA, 91125, USA

    Ben Stevens & Tim Colonius

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  1. Ben Stevens
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  2. Tim Colonius
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Correspondence to Ben Stevens.

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Communicated by Steven L. Brunton.

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This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1745301.

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Stevens, B., Colonius, T. Enhancement of shock-capturing methods via machine learning. Theor. Comput. Fluid Dyn. 34, 483–496 (2020). https://doi.org/10.1007/s00162-020-00531-1

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