发表机构
Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种通过核分解减去格林函数奇异部分来学习椭圆算子解算子的框架，将问题转化为边界上的调和延拓学习，显著提升学习速率，并扩展到半线性问题。
AI 中文摘要
本工作分析了一个用于学习一般自伴二阶椭圆算子 $\cL$ 的解算子（等价地，格林函数）的框架。减去格林函数的奇异部分（该部分要么解析已知，要么由显式的拉普拉斯参数矩阵提供），将学习 $\cL^{-1}$ 重新表述为在边界上学习一个 $\cL$-调和延拓算子。我们将该过程表述为再生核希尔伯特空间（RKHS）之间的算子学习，以表明核分解策略将代数输入速率提高了两个阶次，并且在诱导边界数据具有一致解析正则性以及兼容的输出空间下，将输出速率从代数提升至指数。我们进一步通过迭代分解将该框架扩展到半线性问题，其中唯一学习的对象仍然是线性算子，确保改进的线性速率直接转化为非线性迭代。提供了数值实验以验证理论发展。
英文摘要
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.
