变分问题求解的最优恢复方法
Optimal Recovery for Solving Variational Problems
- Booz Allen Hamilton Inc（博思艾伦咨询公司）
- U.S. Army Combat Capabilities Development Command (DEVCOM) Army Research Laboratory（美国陆军作战能力发展司令部（DEVCOM）陆军研究实验室）
- Drexel University（德雷塞尔大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对变分问题，提出基于再生核希尔伯特空间最优恢复的两步求解框架，结合稀疏Cholesky分解与Γ-收敛理论，实现高效、鲁棒且精确的能量最小化。
AI中文摘要：
在科学与工程中，许多物理定律和科学原理自然地以变分问题的形式出现，即在适当的函数空间上最小化能量泛函。在许多情况下，直接发现能量泛函的最小值点比求解相应的欧拉-拉格朗日方程更为有利。传统的数值求解器，如有限元方法（FEM），往往缺乏融入先验知识或含噪声观测数据的灵活性。近年来，机器学习方法，特别是基于核的方法，在科学计算中引起了越来越多的关注。与需要域离散化和网格生成的FEM不同，基于核的方法从散乱节点构造解，从而避免了网格划分的复杂性，尤其是在高维或几何复杂域中。本文提出了一种基于再生核希尔伯特空间（RKHS）中最优恢复公式的两步变分能量最小化程序，提供了一个统一框架，可无缝地结合物理约束和含噪声数据。在计算方面，我们采用Matérn核的稀疏Cholesky分解来缓解核方法众所周知的立方复杂度瓶颈。在理论方面，我们利用Γ-收敛理论严格建立了所提方法的存在性和收敛性理论。基准问题上的数值实验证明了该方法的效率、鲁棒性和准确性。
英文摘要：
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.
