基于可分离神经算子的函数间回归
Function-On-Function Regression Through Separable Neural Operators
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中文总结 AI 辅助
该研究针对函数间回归算子估计问题，提出可分离神经算子方法并证明其估计量一致性，将其应用于BGC Argo数据以展现海洋学研究潜力。
中文摘要 AI 辅助
本文研究函数间回归模型中回归算子的估计问题。传统研究主要聚焦于线性模型或其直接非线性扩展，而我们提出一种神经算子方法，以在温和的光滑性假设下适配一般回归算子。算子学习是机器学习中新兴的活跃领域，尤其适用于求解由偏微分方程控制的物理模型。基于该范式，我们的方法引入了可分离神经算子，这是一种通过依赖输入的系数函数与依赖输出的基函数来表示回归算子的神经算子架构。除将该架构适配至回归算子估计问题外，我们还在相对温和的光滑性与采样条件下证明了该估计量的一致性，允许函数数据在密集、可能不规则的离散网格上被观测。我们还将所提方法应用于BGC Argo数据，展示其在海洋学研究中的潜力。
英文摘要
This paper investigates the estimation of the regression operator in function-on-function regression models. While traditional research has predominantly focused on linear models or their immediate nonlinear extensions, we propose a neural operator approach to accommodate general regression operators under mild smoothness assumptions. Operator learning has emerged as an active area of machine learning, particularly for solving physical models governed by partial differential equations. Using this paradigm, our methodology introduces the separable neural operator, a neural-operator architecture that represents the regression operator through input-dependent coefficient functions and output-dependent basis functions. Beyond adapting this architecture to the regression operator estimation problem, we establish the consistency of the estimator under relatively mild smoothness and sampling conditions, allowing functional data to be observed on dense, possibly irregular, discrete grids. We also apply the proposed approach to the BGC Argo data and demonstrate its potential for oceanographic research.
