发表机构
Fudan University; University of Sydney（复旦大学; 悉尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出通过极限核分析编码器-解码器算子学习，实现分辨率无关的正则性假设，并给出正则化SGD的误差界及神经网络扩展，涵盖多种核与采样表示。
AI 中文摘要
算子学习是在函数空间上形式化的，但训练数据通常仅通过有限维表示可用。在编码器-解码器架构中，编码空间上的矩阵值核在原始函数空间上诱导出算子值核，相应的再生核希尔伯特空间是等距同构的。随着输入和输出分辨率的增加，诱导核收敛到极限核，其关联积分算子按算子范数收敛，从而允许正则性假设独立于编码分辨率。对于正则化随机梯度下降，我们建立了递减和固定步长的上界，将编码项和正则化项与阶为\\(t^{-\theta}\\)和\\(T^{-\theta'}\\)的优化项分离，其中\\(\theta,\theta'\in(0,1)\\)任意。我们进一步证明了这些编码诱导项通常是不可避免的下界。该分析进一步扩展到编码器-解码器神经网络，通过极限神经正切核（NTK），当编码误差代数衰减时，产生带有额外有限宽度项的误差界以及多项式参数和样本复杂度保证。该框架涵盖了由径向核和点积核构造的矩阵值核、由宽编码器-解码器神经网络产生的NTK，以及基于傅里叶、勒让德多项式、小波、PCA或逐点采样表示的编码器-解码器对。
英文摘要
Operator learning is formulated on function spaces, but training data are typically available only through finite-dimensional representations. In encoder--decoder architectures, a matrix-valued kernel on the encoded space induces an operator-valued kernel on the original function spaces, and the corresponding reproducing kernel Hilbert spaces are isometrically isomorphic. As the input and output resolutions increase, the induced kernels converge to a limiting kernel, in the sense of operator-norm convergence of their associated integral operators, allowing regularity assumptions to be stated independently of the encoding resolution. For regularized stochastic gradient descent, we establish upper bounds for decreasing and fixed step sizes, separating the encoding and regularization terms from optimization terms of order \(t^{-θ}\) and \(T^{-θ'}\), respectively, for any \(θ,θ'\in(0,1)\). We further prove lower bounds showing that these encoding-induced terms are generally unavoidable. The analysis is further extended to encoder--decoder neural networks through the limiting neural tangent kernel (NTK), yielding error bounds with an additional finite-width term and polynomial parameter and sample complexity guarantees when the encoding errors decay algebraically. The framework covers matrix-valued kernels constructed from radial and dot product kernels, NTKs arising from wide encoder--decoder neural networks, and encoder--decoder pairs based on Fourier, Legendre polynomial, wavelet, PCA, or pointwise sampling representations.
Comments69 pages
