Dimensionality reduction of solution reconstruction methods for a four-point stencil

Abstract

The development of reconstruction methods has faced considerable challenges due to their inherent high dimensionality. In the present study, an innovative dimensionality reduction method aimed at mitigating these challenges by normalizing flow variables is proposed. Through our investigation, we demonstrate that a reconstruction method, specifically designed for a four-point stencil that is compatible with unstructured meshes, can be effectively represented by six two-dimensional functions. This key insight enables us to devise a visualization technique utilizing a single contour plot for the reconstruction method. Additionally, we establish that a single data set can adequately represent the reconstruction method, facilitating solution reconstruction through data set interpolation. By carefully evaluating the interpolation error, a data set of reasonable size yields sufficiently small interpolation errors. Notably, we uncover the possibility of extracting reconstruction methods from a trained artificial neural network (ANN). To gauge the impact of accumulated interpolation errors on solution quality, we conduct comprehensive analyses on four benchmark problems. Our results demonstrate that with a data set of sufficient size, the accumulated interpolation error becomes negligible, rendering the solution reconstruction by interpolating the extracted data set both accurate and cost-effective. The implications of our findings hold substantial promise for enhancing the efficiency and efficacy of reconstruction methods.