Improving Learning of Green's Functions Through Kernel Decomposition
Abstract
This work analyzes a framework for learning the solution operator, or equivalently, the Green's function, of a general self-adjoint second-order elliptic operator cL. Subtracting the singular part of the Green's function, either known analytically or supplied by an explicit Laplace parametrix, recasts the learning of cL−1 as learning an cL-harmonic extension operator on the boundary. We formulate the process as operator learning between reproducing kernel Hilbert spaces (RKHSs) to show that the kernel decomposition strategy raises the algebraic input rate by two orders and, under uniform analytic regularity of the induced boundary data together with compatible output spaces, upgrades the output rate from algebraic to exponential. We extend the framework further to semilinear problems via an iterative decomposition in which the only learned object remains a linear operator, ensuring that the improved linear rates translate directly to the nonlinear iteration. Numerical experiments are provided to validate the theoretical development.
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BibTeX
@misc{chen2026improving,
title = {{Improving Learning of Green's Functions Through Kernel Decomposition}},
author = {Beiji Chen and Qiang Du and Kui Ren and Tian-Yi Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09101},
eprint = {2610.09101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09101},
}
RIS
TY - UNPB
TI - Improving Learning of Green's Functions Through Kernel Decomposition
AU - Beiji Chen
AU - Qiang Du
AU - Kui Ren
AU - Tian-Yi Zhou
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09101
UR - https://arxiv.org/abs/2610.09101
AB - This work analyzes a framework for learning the solution operator, or equivalently, the Green's function, of a general self-adjoint second-order elliptic operator cL. Subtracting the singular part of the Green's function, either known analytically or supplied by an explicit Laplace parametrix, recasts the learning of cL^−1 as learning an cL-harmonic extension operator on the boundary. We formulate the process as operator learning between reproducing kernel Hilbert spaces (RKHSs) to show that the kernel decomposition strategy raises the algebraic input rate by two orders and, under uniform analytic regularity of the induced boundary data together with compatible output spaces, upgrades the output rate from algebraic to exponential. We extend the framework further to semilinear problems via an iterative decomposition in which the only learned object remains a linear operator, ensuring that the improved linear rates translate directly to the nonlinear iteration. Numerical experiments are provided to validate the theoretical development.
ER -