Training an artificial neural network to detect discontinuities in solutions of conservation laws

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EPFL mathematicians have developed a new way to detect discontinuities by training an artificial neural network using supervised learning to classify cells as troubled-cells or good-cells. The network is trained offline and can be easily integrated into existing code frameworks.

When an aircraft reaches the speed of sound, a shockwave is formed. If a shockwave carries energy, like an ordinary wave, it is characterized by an abrupt change in the features of the medium, such as pressure or temperature. Due to that brutal transition, shockwaves must be treated as a discontinuous transition.

We know that solutions – the shockwave for example – of special time-dependent partial differential equations (PDEs), known as conservation laws, often develop discontinuities. Another typical example is the shallow water equations describing flow dynamics in water bodies. However, while approximating these discontinuities, numerical algorithms suffer from spurious oscillations that can lead to numerical instabilities, resulting in unphysical results.

A popular technique to overcome this issue is by limiting the numerical solution in problematic cells of the mesh used to decompose the physical domain. However, in order to use this method, one must locate the problematic cells. This is achieved via a suitable troubled-cell indicator, an algorithm that flags the cells in which the solution is unstable. Although several such indicators have been developed over the years, most of them need problem-dependent parameters to be prescribed. A non-optimal choice of these parameters can either lead to the reappearance of the oscillations or the loss of accuracy in regions of the domain where the solution is smooth.

Professor Jan S. Hesthaven, head of the Chair of Computational Mathematics and Simulation Science (MCSS), and his postdoc Dr. Deep Ray are investigating the effectiveness of deep-learning techniques to resolve issues of parameter-tuning when computationally solving partial differential equations. Their research has been published in the Journal of Computational Physics.

Representation of density in a solution of the Euler equation of gas dynamics.

Overcoming algorithmic bottlenecks with deep neural networks

Their new approach for detecting discontinuities involves training an artificial neural network, during an offline stage, with data generated from simple canonical functions. The advantage of this strategy is that it is parameter-free, computationally efficient, problem independent, and can be integrated into existing code frameworks. Furthermore, the network was demonstrated to outperform traditional troubled-cell indicators. Combining these properties, the network is attractive as a universal troubled-cell indicator for general conservation laws.

Deep learning has great potential to solve problems which arise in the computational sciences, and there has recently been an active interest in the use of such networks. However, the philosophy of Hesthaven and his team is that deep learning should not completely replace existing, well-established numerical methods but should rather support them to enhance their performance.

The EPFL mathematicians are now working on suppressing the oscillations by introducing an artificial viscous term in the underlying partial differential equation. Still aiming to make their method parameter-free, they have trained neural networks to predict the required viscosity. The results are promising, as the networks beat existing methods on both structured and unstructured meshes.

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