(untitled) (Communications in Computational Physics, 2025)
In this paper, the model reduction method based on k-nearest-neighbors is provided for the parametrized nonlocal partial differential equations (PDEs). In comparison to standard local PDEs, the stiffness matrix of the corresponding nonlocal model loses sparsity due to the nonlocal interaction parameter δ. Specially the nonlocal model contains uncertain parameters, enhancing the complexity of computation. In order to improve the computation efficiency, we combine the k-nearest-neighbors with the model reduction method to construct the efficient surrogate models of the parametrized nonlocal problems. This method is an offline-online mechanism. In the offline phase, we develop the full-order model by using the quadratic finite element method (FEM) to generate snapshots and employ the model reduction method to process the snapshots and extract their key characters. In the online phase, we utilize k-nearest-neighbors regression to construct the surrogate model. In the numerical experiments, we first verify the convergence rate when applying quadratic FEM to the nonlocal problems. Subsequently, for the linear and nonlinear nonlocal problems with random inputs, the numerical results illustrate the efficiency and accuracy of the surrogate models.