Kernel Principal Component Analysis for Allen-Cahn Equations

Different researchers have analyzed effective computational methods that maintain the precision of Allen-Cahn (AC) equations and their constant security. This article presents a method known as the reduced-order model technique by utilizing kernel principle component analysis (KPCA), a nonlinear variation of traditional principal component analysis (PCA). KPCA is utilized on the data matrix created using discrete solution vectors of the AC equation. In order to achieve discrete solutions, small variations are applied for dividing up extraterrestrial elements, while Kahan's method is used for temporal calculations. Handling the process of backmapping from small-scale space involves utilizing a non-iterative formula rooted in the concept of the multidimensional scaling (MDS) method. Using KPCA, we show that simplified sorting methods preserve the dissipation of the energy structure. The effectiveness of simplified solutions from linear PCA and KPCA, the retention of invariants, and computational speeds are shown through one-, two-, and three-dimensional AC equations.