Deep Operator Networks are an increasingly popular paradigm for solving regression in infinite dimensions and hence solve families of PDEs in one shot. In this work, we aim to establish a first-of-its-kind data-dependent lowerbound on the size of DeepONets required for them to be able to reduce empirical error on noisy data. In particular, we show that for low training errors to be obtained on $n$ data points it is necessary that the common output dimension of the branch and the trunk net be scaling as $\Omega \left ( \sqrt[\leftroot{-1}\uproot{-1}4]{n} \right )$. This inspires our experiments with DeepONets solving the advection-diffusion-reaction PDE, where we demonstrate the possibility that at a fixed model size, to leverage increase in this common output dimension and get monotonic lowering of training error, the size of the training data might necessarily need to scale at least quadratically with it.

翻译：深度算子网络是解决无限维回归问题并一次性求解偏微分方程族的一种日益流行的范式。本文旨在建立首个与数据相关的下界，研究深度算子网络在含噪数据上降低经验误差所需的规模。特别地，我们证明：要在$n$个数据点上获得较低的训练误差，分支网络与主干网络的公共输出维度必须满足$\Omega \left ( \sqrt[\leftroot{-1}\uproot{-1}4]{n} \right )$的尺度。这启发我们利用深度算子网络求解对流-扩散-反应偏微分方程的实验，其中我们展示了在固定模型规模下，若想通过增加该公共输出维度来单调降低训练误差，训练数据的规模可能至少需要与该维度呈二次缩放关系。