Nonlinear Optimal Recovery in Hilbert Spaces

2025/05/31 by Qiang Du, Lin, Daozhe, Du, Qiang
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.00704

openalex publication_date 2025/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear optimal recovery problem, establishing its well-posedness and proving its convergence to the true solution as the number of measurements increases. However, the resulting formulation might not have a finite-dimensional solution in general. We thus present a sufficient condition for the finite dimensionality of the solution, applicable to problems with well-defined point evaluation measurements. To address the broader setting, we introduce a relaxed nonlinear optimal recovery and provide a detailed convergence analysis. An illustrative example is given to demonstrate that our formulations and theoretical findings offer a comprehensive framework for solving nonlinear problems in infinite-dimensional spaces with limited data.

Citations

• The Parametric Complexity of Operator Learning
• Sparse Representer Theorems for Learning in Reproducing Kernel Banach Spaces
• Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs
• Solving PDEs with Incomplete Information
• NOMAD: Nonlinear Manifold Decoders for Operator Learning
• Solving and Learning Nonlinear PDEs with Gaussian Processes
• Representer Theorems in Banach Spaces: Minimum Norm Interpolation,\n Regularized Learning and Semi-Discrete Inverse Problems
• Model Reduction and Neural Networks for Parametric PDEs
• Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
• A unifying representer theorem for inverse problems and machine learning
• The representer theorem for Hilbert spaces: a necessary and sufficient condition

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