Franco Nicola Rares, Fresca Stefania, Manzoni Andrea, Zunino Paolo
MOX, Math Department, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy.
Neural Netw. 2023 Apr;161:129-141. doi: 10.1016/j.neunet.2023.01.029. Epub 2023 Jan 26.
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 μ∈R onto a functional output u:[0,1]→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.
最近，深度卷积神经网络（CNNs）在诸如参数化偏微分方程的降阶建模等领域得到应用时已被证明是成功的。尽管它们具有准确性和效率，但文献中现有的方法在其数学基础上仍缺乏严格的论证。受这一事实的启发，在本文中，我们通过CNN模型推导了非线性算子逼近的严格误差界。更确切地说，我们处理的情况是，一个算子将有限维输入μ∈R映射到一个泛函输出u：[0,1]→R，并且使用神经网络模型来逼近输入到输出映射的离散化版本。所得的误差估计为定义神经网络架构的超参数提供了清晰的解释。所有证明都是构造性的，并且最终揭示了CNNs与傅里叶变换之间的深刻联系。最后，我们通过数值实验对推导的误差界进行补充，以说明它们的应用。
