Abstract
Among several recently proposed data-driven Reduced Order Models (ROMs), deep learning-based ROMs (DL-ROMs) have proven to be a successful strategy for constructing 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 of 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 of (i) incorporating knowledge of physical laws, (ii) handling varying geometries, (iii) identifying the latent dynamics to ensure accurate out-of-training forecasts, and (iv) including uncertainty quantification.
In all these cases, we will demonstrate how the construction of a suitably expressive—and potentially explainable—latent space is essential to ensure the accuracy and efficiency of reduced-order models that utilize deep neural networks, while also drawing conclusions that may be of interest in other contexts within scientific machine learning.
