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 $\Omega \left ( \sqrt[\leftroot{-1}\uproot{-1}6]{n} \right )$. 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：深度算子网络是一种日益流行的范式，用于解决无限维空间中的回归问题，从而一次性求解一族偏微分方程。本文旨在建立首个基于数据的DeepONet规模下界，该下界要求网络能够降低含噪数据上的经验误差。特别地，我们证明要在$n$个数据点上获得低训练误差，分支网络和主干网络的公共输出维度必须满足$\Omega \left ( \sqrt[\leftroot{-1}\uproot{-1}6]{n} \right )$的量级。这启发了我们利用DeepONet求解对流-扩散-反应偏微分方程的实验，其中我们展示了如下可能性：在固定模型规模下，若想利用公共输出维度的增加并实现训练误差的单调降低，训练数据的规模可能必须至少与该维度呈二次方比例缩放。