Why is the cylinder flow a terrible test case for deep learning?

Recently, deep learning has attracted a lot attention and its successes are regularly reported by both the scientific and mainstream media. Within the fluid dynamics community, the two-dimensional cylinder flow is often used as a test case to illustrate the performances of different network architectures for tasks such as reduced-order modeling, flow field estimation or nonlinear control. Despite its wide use as a representative test case for "complex" nonlinear dynamics, the inherent low-dimensionality of this flow can be captured by a fairly simple model. The reduced-order model proposed herein not only mimics the nonlinear dynamics of the system but also accounts for the mode deformation that occurs as the flow evolves from the base flow to the mean flow. Based on ideas from dynamical systems and differential geometry, the simplicity and accuracy of our data-driven model provide hints about which features recently proposed neural network models may have actually learned. Its simplicity moreover strongly underlines that the aforementioned neural networks are likely to be overly complex for the configuration considered and that, as a consequence, the performances reported for the cylinder flow may not be indicative of what would be obtained for more realistic configurations.