提出变分空间理论，实现高效神经算子逼近。
Efficient Approximation for Encoder--Decoder Neural Operators via Variation Spaces
- 基于函数空间理论构建变分空间，定义输入输出测度结构
- 两层网络误差分解为编码误差与$N^{-1/2}$阶有限宽项
- 适用于高维输入输出的高效算子学习，理论保障更强
我们研究使用编码器-解码器神经网络进行算子学习。受神经网络函数空间理论启发，引入变分空间作为非线性算子的无限维结构类，通过直接定义在输入输出空间上的向量值测度实现。对于该空间中的算子，我们在Bochner $L^q$范数下建立了编码器-解码器两层网络的逼近界。误差界分解为输入编码误差、输出编码误差及$N^{-1/2}$阶有限宽度逼近项，常数不依赖编码维数。当编码误差随编码维数多项式衰减时，可得代数逼近与学习率。结果为超越一般Lipschitz或Fréchet可微算子类的高效神经算子学习提供了理论保证。
原文摘要 · Abstract (English)
We study operator learning using encoder--decoder neural networks. Inspired by the function-space theory of neural networks, we introduce a variation space as an infinite-dimensional structural class for nonlinear operators. This space is defined through vector-valued measures directly on the input and output spaces. For operators in this space, we establish approximation bounds for encoder--decoder two-layer networks in the Bochner $L^q$ norm. The resulting error bound decomposes into the input encoding error, the output encoding error, and a finite-width approximation term of order $N^{-1/2}$, with a constant independent of the input and output encoding dimensions. When the input and output encoding errors decay polynomially in the encoding dimensions, these estimates yield algebraic approximation and learning rates. The results provide an theoretical guarantees for efficient neural operator learning beyond general Lipschitz or Fréchet differentiable operator classes.
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