Nonparametric learning of kernels in nonlocal operators
This addresses the challenge of learning kernels in nonlocal operators for applications like stress wave propagation in heterogeneous solids, offering a robust solution for inverse problems in scientific computing, though it is incremental as it builds on existing regularization methods.
The paper tackles the ill-posed inverse problem of learning kernels in nonlocal operators, which leads to divergent estimators under errors or noise, and proposes a nonparametric regression algorithm with data adaptive RKHS Tikhonov regularization to yield a noisy-robust convergent estimator, outperforming baselines in robustness, generalizability, and accuracy on synthetic and real-world datasets.
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.