An Operator Learning Approach via Function-valued Reproducing Kernel\n Hilbert Space for Differential Equations

2022/02/18 by Kaijun Bao, Bao, Kaijun, Qian, Xu +5
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2202.09488

openalex publication_date 2022/02/18 · openalex created_date 2022/04/26 · openalex updated_date 2026/07/28

Abstract

Much recent work has addressed the solution of a family of partial\ndifferential equations by computing the inverse operator map between the input\nand solution space. Toward this end, we incorporate function-valued reproducing\nkernel Hilbert spaces in our operator learning model. We use neural networks to\nparameterize Hilbert-Schmidt integral operator and propose an architecture.\nExperiments including several typical datasets show that the proposed\narchitecture has desirable accuracy on linear and nonlinear partial\ndifferential equations even with a small amount of data. By learning the\nmappings between function spaces, the proposed method can find the solution of\na high-resolution input after learning from lower-resolution data.\n

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