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.

翻译：算子学习问题出现在科学计算的许多关键领域，其中偏微分方程被用于建模物理系统。在此类场景中，算子映射于巴拿赫空间或希尔伯特空间之间。本工作致力于解决巴拿赫空间间的算子学习问题，这与过去绝大多数仅考虑希尔伯特空间的研究形成对比。我们专注于学习全纯算子——这是一类具有广泛应用的重要问题。我们将任意近似编码器与解码器，与标准前馈深度神经网络架构（具体而言，宽度恒定且超过深度的架构）相结合，并在标准$\ell^2$损失最小化框架下进行研究。我们首先确定了一族深度神经网络，使得由此产生的深度学习过程对此类算子达到最优泛化界。对于标准全连接架构，我们随后证明训练问题存在不可数多个极小值点，它们能产生等效的最优性能。我们所考虑的深度神经网络架构是“问题无关的”，其宽度和深度仅取决于训练数据量$m$，而不依赖于目标算子的正则性假设。接着，我们证明深度学习对此问题是最优的：任何恢复方法都无法超越这些泛化界（至多相差对数项）。最后，我们展示了数值结果，以证明该方法在具有挑战性问题上的实际性能，包括参数化扩散、Navier-Stokes-Brinkman和Boussinesq偏微分方程。