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POD-DL-ROMs have been recently proposed as an extremely versatile strategy tobuild accurate and reliable reduced order models (ROMs) for nonlinearparametrized partial differential equations, combining (i) a preliminarydimensionality reduction obtained through proper orthogonal decomposition (POD)for the sake of efficiency, (ii) an autoencoder architecture that furtherreduces the dimensionality of the POD space to a handful of latent coordinates,and (iii) a dense neural network to learn the map that describes the dynamicsof the latent coordinates as a function of the input parameters and the timevariable. Within this work, we aim at justifying the outstanding approximationcapabilities of POD-DL-ROMs by means of a thorough error analysis, showing howthe sampling required to generate training data, the dimension of the PODspace, and the complexity of the underlying neural networks, impact on thesolution accuracy. This decomposition, combined with the constructive nature ofthe proofs, allows us to formulate practical criteria to control the relativeerror in the approximation of the solution field of interest, and derivegeneral error estimates. Furthermore, we show that, from a theoretical point ofview, POD-DL-ROMs outperform several deep learning-based techniques in terms ofmodel complexity. Finally, we validate our findings by means of suitablenumerical experiments, ranging from parameter-dependent operators analyticallydefined to several parametrized PDEs.

Error estimates for POD-DL-ROMs: a deep learning framework for reduced  order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal  decomposition