Generalized MLS and LS-SVR for Allen-Cahn Equations on Spheres
This work extends least-squares support vector regression (LS-SVR) based on the generalized moving least squares (GMLS) approximation to the numerical solution of partial differential equations on spheres. We show that the proposed approach constructs a kernel-based approximant in its dual form and establishes its connection with the associated reproducing kernel Hilbert space (RKHS). Furthermore, solving the PDE through the proposed LS-SVR framework is shown to be equivalent to solving a smoothing problem or a kernel ridge regression problem in the corresponding RKHS. As applications, the method is employed to solve the Allen-Cahn equation and its conservative counterpart on spheres. An implicit first-order operator-splitting scheme is used to separate each model into linear and nonlinear subproblems. The linear subproblem is discretized by the backward differentiation formula and approximated at each time step using LS-SVR derived from a GMLS-based collocation formulation, while the nonlinear part is treated analytically. The resulting scheme is shown to admit a unique numerical solution at every time step. For the conservative Allen-Cahn model, a correction procedure is incorporated to preserve mass. Numerical experiments demonstrate the accuracy and effectiveness of the proposed framework.