A kernel ridge regression combining nonlinear ROMs for accurate flow-field reconstruction with discontinuities

Manifold learning (ML), one of the most representative nonlinear reduced-order models (ROMs), can effectively capture the nonlinear flow characteristics of the entire flow fields. However, the reconstruction from low-dimensional manifold coordinates to high-dimensional flow fields often introduces considerable reconstruction errors, leading to inaccurate reconstruction in the location with nonlinear flow structures, especially the region with discontinuities. To address this challenge, a novel reconstruction method based on nonlinear ROMs is proposed to enhance the accuracy of reconstructing flow fields with discontinuities. Inspired by the proper orthogonal decomposition (POD), we introduce a kernel function to obtain the mode coefficients in the nonlinear kernel space, and perform ridge regression to construct a set of modes that effectively capture the discontinuity features present in the flow fields. Then, we combine these coefficients and modes to achieve accurate reconstruction of flow fields with discontinuities. The proposed flow-field reconstruction method is validated through the reconstruction of transonic flow fields over the RAE2822 airfoil. Comparison results demonstrate that the method can achieve better reconstruction accuracy than the existing approaches. In comparison with POD, the modes obtained via the kernel ridge regression (KRR) appear to capture the local discontinuities more precisely. This work provides an effective and highly interpretable approach for enhancing the accuracy of nonlinear ROMs in the modeling of discontinuous flow fields.