论文标题
操作员学习中卷积神经网络的近似范围
Approximation bounds for convolutional neural networks in operator learning
论文作者
论文摘要
最近，在诸如参数化PDES的订购订单模型降低时，深度卷积神经网络（CNN）已被证明是成功的。尽管它们的准确性和效率，但文献中可用的方法仍然缺乏其数学基础的严格理由。在这一事实中，在本文中，我们通过CNN模型得出了严格的误差界限，以实现非线性操作员的近似值。更准确地说，我们解决了操作员将有限的尺寸输入$ \boldsymbolμ\ iN \ mathbb {r}^{p} $映射到功能输出$ _ {\boldsymbolμ}上的情况下输入到输出图。由此产生的错误估计值清楚地解释了定义神经网络体系结构的超参数。所有证明都是建设性的，它们最终揭示了CNN与傅立叶变换之间的深厚联系。最后，我们通过说明其应用的数值实验来补充派生的误差界限。
Recently, deep Convolutional Neural Networks (CNNs) have proven to be successful when employed in areas such as reduced order modeling of parametrized PDEs. Despite their accuracy and efficiency, the approaches available in the literature still lack a rigorous justification on their mathematical foundations. Motivated by this fact, in this paper we derive rigorous error bounds for the approximation of nonlinear operators by means of CNN models. More precisely, we address the case in which an operator maps a finite dimensional input $\boldsymbolμ\in\mathbb{R}^{p}$ onto a functional output $u_{\boldsymbolμ}:[0,1]^{d}\to\mathbb{R}$, and a neural network model is used to approximate a discretized version of the input-to-output map. The resulting error estimates provide a clear interpretation of the hyperparameters defining the neural network architecture. All the proofs are constructive, and they ultimately reveal a deep connection between CNNs and the Fourier transform. Finally, we complement the derived error bounds by numerical experiments that illustrate their application.
