# A Neural Network based Shock Detection and Localization Approach for Discontinuous Galerkin Methods

arXiv: https://arxiv.org/abs/2001.08201 · HTML: https://mlsanity.com/paper/2001.08201/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jan 2020.

## Built on by (most cited first, 8 of 8)

- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [Deep Reinforcement Learning for Computational Fluid Dynamics on HPC Systems](https://mlsanity.com/paper/2205.06502/) (2022)
- [Toward Discretization-Consistent Closure Schemes for Large Eddy Simulation Using Reinforcement Learning](https://mlsanity.com/paper/2309.06260/) (2023)
- [Cell-average based neural network method for hyperbolic and parabolic partial differential equations](https://mlsanity.com/paper/2107.00813/) (2021)
- [Learning WENO for entropy stable schemes to solve conservation laws](https://mlsanity.com/paper/2403.14848/) (2024)
- [An unsupervised machine-learning-based shock sensor for high-order supersonic flow solvers](https://mlsanity.com/paper/2308.00086/) (2023)
- [Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions](https://mlsanity.com/paper/2605.19823/) (2026)
- [Hybrid Neural World Models](https://mlsanity.com/paper/2605.28317/) (2026)

## Abstract (arXiv)

The stable and accurate approximation of discontinuities such as shocks on a finite computational mesh is a challenging task. Detection of shocks or strong discontinuities in the flow solution is typically achieved through a priori troubled cell indicators, which guide the subsequent action of an appropriate shock capturing mechanism. Arriving at a stable and accurate solution often requires empirically based parameter tuning and adjustments of the indicator settings to the discretization and solution at hand. In this work, we propose to separate the task of shock detection and shock capturing more strongly and aim to develop a shock indicator that is robust, accurate, requires minimal user input and is suitable for high order element-based methods like discontinuous Galerkin and flux reconstruction methods. The novel indicator is learned from analytical data through a supervised learning strategy; its input is given by the high order solution field, its output is an element-local map of the shock position. We use state of the art methods from edge detection in image analysis based on deep convolutional multiscale networks and deep supervision to train the indicators. The resulting networks are then used as black box indicators, showing their robustness and accuracy on well established canonical testcases. All simulations are run ab initio using the developed indicators, showing that they provide also stability during the strongly transient phases. In particular for high order schemes with large cells and considerable inner-cell resolution capabilities, we demonstrate how the additional accurate prediction of the position of the shock front can be exploited to guide inner-element shock capturing strategies.