Operator learning problems arise in many key areas of scientific computing
where Partial Differential Equations (PDEs) are used to model physical systems.
In such scenarios, the operators map between Banach or Hilbert spaces. In this
work, we tackle the problem of learning operators between Banach spaces, in
contrast to the vast majority of past works considering only Hilbert spaces. We
focus on learning holomorphic operators - an important class of problems with
many applications. We combine arbitrary approximate encoders and decoders with
standard feedforward Deep Neural Network (DNN) architectures - specifically,
those with constant width exceeding the depth - under standard $\ell^2$-loss
minimization. We first identify a family of DNNs such that the resulting Deep
Learning (DL) procedure achieves optimal generalization bounds for such
operators. For standard fully-connected architectures, we then show that there
are uncountably many minimizers of the training problem that yield equivalent
optimal performance. The DNN architectures we consider are `problem agnostic',
with width and depth only depending on the amount of training data $m$ and not
on regularity assumptions of the target operator. Next, we show that DL is
optimal for this problem: no recovery procedure can surpass these
generalization bounds up to log terms. Finally, we present numerical results
demonstrating the practical performance on challenging problems including the
parametric diffusion, Navier-Stokes-Brinkman and Boussinesq PDEs.

通过将任意近似编码器和解码器与标准前馈深度神经网络 (DNN) 体系结构相结合，我们提出了学习巴拿赫空间之间的算子的问题。我们首先确定了一组 DNN 的族群，使得由这些深度学习 (DL) 过程所获得的产生出算子可以达到最佳的泛化性能。接下来，我们证明了 DL 对于这个问题是最优的，没有任何恢复过程可以超越这些泛化界限。最后，我们展示了在具有挑战性的问题上的实际性能，包括参数扩散、Navier-Stokes-Brinkman 和 Boussinesq 偏微分方程组。