Nonparametric learning of kernels in nonlocal operators
- URL: http://arxiv.org/abs/2205.11006v1
- Date: Mon, 23 May 2022 02:47:55 GMT
- Title: Nonparametric learning of kernels in nonlocal operators
- Authors: Fei Lu, Qingci An, Yue Yu
- Abstract summary: We provide a rigorous identifiability analysis and convergence study for the learning of kernels in nonlocal operators.
We propose a nonparametric regression algorithm with a novel data adaptive RKHS Tikhonov regularization method based on the function space of identifiability.
- Score: 6.314604944530131
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Nonlocal operators with integral kernels have become a popular tool for
designing solution maps between function spaces, due to their efficiency in
representing long-range dependence and the attractive feature of being
resolution-invariant. In this work, we provide a rigorous identifiability
analysis and convergence study for the learning of kernels in nonlocal
operators. It is found that the kernel learning is an ill-posed or even
ill-defined inverse problem, leading to divergent estimators in the presence of
modeling errors or measurement noises. To resolve this issue, we propose a
nonparametric regression algorithm with a novel data adaptive RKHS Tikhonov
regularization method based on the function space of identifiability. The
method yields a noisy-robust convergent estimator of the kernel as the data
resolution refines, on both synthetic and real-world datasets. In particular,
the method successfully learns a homogenized model for the stress wave
propagation in a heterogeneous solid, revealing the unknown governing laws from
real-world data at microscale. Our regularization method outperforms baseline
methods in robustness, generalizability and accuracy.
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