We establish universality and expression rate bounds for a class of neural Deep Operator Networks (DON) emulating Lipschitz (or H\"older) continuous maps $\mathcal G:\mathcal X\to\mathcal Y$ between (subsets of) separable Hilbert spaces $\mathcal X$, $\mathcal Y$. The DON architecture considered uses linear encoders $\mathcal E$ and decoders $\mathcal D$ via (biorthogonal) Riesz bases of $\mathcal X$, $\mathcal Y$, and an approximator network of an infinite-dimensional, parametric coordinate map that is Lipschitz continuous on the sequence space $\ell^2(\mathbb N)$. Unlike previous works ([Herrmann, Schwab and Zech: Neural and Spectral operator surrogates: construction and expression rate bounds, SAM Report, 2022], [Marcati and Schwab: Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations, SAM Report, 2022]), which required for example $\mathcal G$ to be holomorphic, the present expression rate results require mere Lipschitz (or H\"older) continuity of $\mathcal G$. Key in the proof of the present expression rate bounds is the use of either super-expressive activations (e.g. [Yarotski: Elementary superexpressive activations, Int. Conf. on ML, 2021], [Shen, Yang and Zhang: Neural network approximation: Three hidden layers are enough, Neural Networks, 2021], and the references there) which are inspired by the Kolmogorov superposition theorem, or of nonstandard NN architectures with standard (ReLU) activations as recently proposed in [Zhang, Shen and Yang: Neural Network Architecture Beyond Width and Depth, Adv. in Neural Inf. Proc. Sys., 2022]. We illustrate the abstract results by approximation rate bounds for emulation of a) solution operators for parametric elliptic variational inequalities, and b) Lipschitz maps of Hilbert-Schmidt operators.

我们建立了一类神经深度操作符网络（DON）的普遍性和表达速度上界，这些网络模拟了（子集）可分离希尔伯特空间 X 到 Y 之间的Lipschitz（或Hölder）连续映射 G。与之前的工作不同，这些表达速率结果仅需G的 Lipschitz（或Hölder）连续性，并且证明这些表达速率界的关键在于使用超表达激活或非标准的包含标准（ReLU）激活函数的神经网络架构。我们通过近似速率界来说明这些抽象结果，其中包括参数椭圆变分不等式的解算符以及希尔伯特-施密特算子的Lipschitz映射。

深层运算网络对于Lipschitz算子的逼近速率