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Recently, deep Convolutional Neural Networks (CNNs) have proven to besuccessful when employed in areas such as reduced order modeling ofparametrized PDEs. Despite their accuracy and efficiency, the approachesavailable in the literature still lack a rigorous justification on theirmathematical foundations. Motivated by this fact, in this paper we deriverigorous error bounds for the approximation of nonlinear operators by means ofCNN models. More precisely, we address the case in which an operator maps afinite dimensional input $\boldsymbol{\mu}\in\mathbb{R}^{p}$ onto a functionaloutput $u_{\boldsymbol{\mu}}:[0,1]^{d}\to\mathbb{R}$, and a neural networkmodel is used to approximate a discretized version of the input-to-output map.The resulting error estimates provide a clear interpretation of thehyperparameters defining the neural network architecture. All the proofs areconstructive, and they ultimately reveal a deep connection between CNNs and theFourier transform. Finally, we complement the derived error bounds by numericalexperiments that illustrate their application.

Approximation bounds for convolutional neural networks in operator learning