Optimal Recovery for Solving Variational Problems

Abstract

In science and engineering, many physical laws and scientific principles naturally arise as variational problems of minimizing an energy functional over an appropriate functional space. In many cases, it is more advantageous to directly discover the minimizer of the energy functional than to solve the associated Euler-Lagrange equations. Conventional numerical solvers, such as the finite-element method (FEM), often lack the flexibility to incorporate prior knowledge or noisy observational data. In recent years, machine learning approaches, particularly kernel-based methods, have attracted growing attention in scientific computing. Unlike FEM, which requires domain discretization and mesh generation, kernel-based methods construct solutions from scattered nodes, thereby avoiding the complexity of meshing, especially in high-dimensional or geometrically complex domains. This works presents a two-step procedure for variational energy minimization based on the optimal recovery formulation in a reproducing kernel Hilbert space (RKHS), providing a unified framework for seamlessly incorporating both physical constraints and noisy data. On the computational side, we employ the sparse Cholesky decomposition for Matérn kernels to alleviate the well-known cubic complexity bottleneck of kernel methods. On the theoretical side, we rigorously establish the existence and convergence theory of the proposed method using $Γ$-convergence theory. Numerical experiments on benchmark problems demonstrate the efficiency, robustness and accuracy of the approach.