Résumé
Among several recently proposed data-driven Reduced Order Models (ROMs), deep learning-based ROMs (DL-ROMs) have proved to be a successful strategy to construct non-intrusive, highly accurate surrogates for the real time solution of parametric nonlinear time-dependent PDEs. By relying on (possibly, convolutional) autoencoders, it is indeed possible to generate latent spaces where the candidate solution is then sought, as a function of parameters and time, using an additional neural network.
In this talk I will provide an overview on DL-ROMs, discussing some recent theoretical results that justify their construction, and connecting them to classical reduced basis methods. Then, I will showcase a series of possible extensions of DL-ROMs capable to (i) handle knowledge of physical laws, (ii) deal with varying geometries, (iii) identify the latent dynamics to ensure accurate out-of-training forecasts, and (iv) include uncertainty quantification.
In all these cases, we will show how the construction of a suitably expressive—and possibly explainable—latent space is essential to ensure accuracy and efficiency of reduced order models exploiting deep neural networks, drawing also some conclusions of possible interest to other contexts in scientific machine learning.
