Deep learning-based reduced order models: recent results and future perspectives

Solving differential problems using full order models (FOMs), like the finite element method, incurs prohibitively computational costs in real-time simulations and multi-query routines. Reduced order modeling aims at replacing FOMs with reduced order models (ROMs), that exhibit significantly reduced complexity while retaining the ability to capture the essential physical characteristics of the system.
Traditional ROMs, rooted in the assumption of linear modal superimposition, such as proper orthogonal decomposition (POD), can prove inadequate when addressing nonlinear time-dependent parametrized partial differential equations (PDEs), especially for scenarios involving coherent structures evolving over time. In response, an alternative approach is proposed, based on deep learning (DL) algorithms, leveraging tools like convolutional neural networks (CNNs), to construct an efficient nonlinear surrogate. In the resulting DL-ROM both the nonlinear trial manifold and the nonlinear reduced dynamics are learned non-intrusively through DL models trained on a dataset of FOM snapshots corresponding to different parameter values. Accuracy and efficiency of the DL-ROM technique are evaluated across various applications, demonstrating the capability to perform real-time computations for new queries.
After presenting some recent achievements in this field, a quick overview of the ongoing research, sketching a list of open issues of potential interest, will be provided.