构建向量值核空间理论，解释神经网络与神经算子的函数结构。
Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators
- 提出向量值再生核巴拿赫空间对，无需限制条件即可建模多输出网络。
- 证明浅层网络和DeepONet、超网络均属于特定积分型向量值核空间。
- 给出泛化表示定理，揭示优化过程可还原实际神经网络结构。
近年来，研究者日益关注神经网络所依赖的函数空间。尽管浅层与深层标量神经网络已被关联至标量再生核巴拿赫空间（RKBS），但$/mathbb{R}^d$-值神经网络与神经算子模型在该框架下的理解仍不充分。为填补这一空白，本文提出向量值再生核巴拿赫空间对（vv-RKBS）的概念，其内在包含一个再生核，并证明每个vv-RKBS必属于此类对。该构造避免了对称核域、有限维输出空间、自反性或可分性等限制性假设，同时保留了向量值再生核希尔伯特空间（vv-RKHS）的熟悉性质。进一步证明，浅层$/mathbb{R}^d$-值神经网络是特定积分型vv-RKBS中的元素；分析DeepONet与超网络架构后，也证实二者同样属于积分型vv-RKBS。在所有情形下，均建立了表示定理，表明在这些函数空间上的优化可恢复对应神经架构。
原文摘要 · Abstract (English)
Recently, there has been growing interest in characterizing the function spaces underlying neural networks. While shallow and deep scalar-valued neural networks have been linked to scalar-valued reproducing kernel Banach spaces (RKBS), $\mathbb{R}^d$-valued neural networks and neural operator models remain less understood in the RKBS setting. To address this gap, we develop a notion of adjoint pairs of vector-valued RKBSs (vv-RKBS), which inherently involves an associated reproducing kernel, and prove that every vv-RKBS belongs to such a pair. Our construction extends existing kernel definitions by avoiding restrictive assumptions such as symmetric kernel domains, finite-dimensional output spaces, reflexivity, or separability, while still recovering familiar properties of vector-valued reproducing kernel Hilbert spaces (vv-RKHS). We then show that shallow $\mathbb{R}^d$-valued neural networks are elements of a specific vv-RKBS, namely an instance of an integral vv-RKBS. To also explore the functional structure of neural operators, we analyze the DeepONet and Hypernetwork architectures and demonstrate that they too belong to an integral vv-RKBS. In all cases, we establish a representer theorem, showing that optimization over these function spaces recovers the corresponding neural architectures.
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