Symbolic regression inspired compressible flow field classification through local characteristic distribution

The characteristic curve (characteristic) serves as a powerful tool for predicting the temporal evolution of solutions to conservation laws. This paper proposes a framework for classifying compressible flow fields from the perspective of the local characteristic distribution of fluid governing equations. First, the Symbolic Regression (SR) method is employed to derive a 2D vortex indicator from data. Inspired by the SR result, the eigenvalue configurations of the gradient matrix of the characteristic eigenvalues along orthogonal directions in mathematics are found to exhibit a one-to-one correspondence with the localized flow structures in physics. Based on this discovery and analysis, this paper pioneers a unified theoretical framework for compressible flow field classification based on local characteristic distribution. The framework is generalizable to 3D scenario and enables the simultaneous identification of vortices, shock waves, expansion waves, and departure structures within a unified theoretical framework, with only a slight increase in computational cost. Extensive 2D and 3D numerical results demonstrate that compared to other classical methods, the framework shows superior robustness against noise, and is capable of detecting dissipative structures, such as weak shocks and small-scale vortices. Classifying flow fields from the perspective of local characteristic distribution offers valuable insights in comprehending the generation, existence, and dissipation mechanisms of flow structures governed by fluid equations, such as shocks and vortices.