证明了神经算子可由有限维编码解码实现，无需额外空间假设。
On finite-dimensional encoding/decoding theorems for neural operators
- 任意局部凸空间间的连续映射可用有限维中间空间逼近。
- 对光滑映射的类似结论仅在源空间具逼近性质时成立。
- 适用于微分方程等非赋范函数空间场景，理论基础更广。
近年来，具有无限维仿射算子的神经网络（神经算子）被用于学习微分方程的解。为实现实际计算，常采用有限维编码/解码定理：任一从函数空间E到F的连续映射，可在紧致一致收敛拓扑下被通过两个有限维巴拿赫空间的连续映射逼近。已有结果（Kovachki等，2023）要求E、F具有逼近性质。本文指出该结论无需任何空间假设，不仅对所有赋范空间成立，也适用于任意局部凸空间。同时，对于C^k光滑映射及C^k紧开拓扑（k≥1），此类逼近成立当且仅当源空间E具有逼近性质。这一分析具有重要意义，因为微分方程理论中常见非赋范的局部凸函数空间，而这些正是神经算子的主要应用领域。
原文摘要 · Abstract (English)
Recently, versions of neural networks with infinite-dimensional affine operators inside the computational units (``neural operator'' networks) have been applied to learn solutions to differential equations. To enable practical computations, one employs finite-dimensional encoding/decoding theorems of the following kind: every continuous mapping $f$ between function spaces $E$ and $F$ is approximated in the topology of uniform convergence on compacta by continuous mappings factoring through two finite dimensional Banach spaces. Such a result is known (Kovachki et al., 2023) for $E,F$ being Banach spaces having the approximation property. We point out that the result needs no assumptions on $E,F$ whatsoever and remains true not only for all normed spaces, but for arbitrary locally convex spaces as well. At the same time, an analogous result for $C^k$-smooth mappings and the $C^k$ compact open topology, $k\geq 1$, holds if and only if the space $E$ has the approximation property. This analysis may be useful already because non-normable locally convex function spaces are common in the theory of differential equations, the main field of applications for the emerging theory.
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