Injecting physics into deep learning based reduced order models: methods for exact mass conservation
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In recent years, deep learning has gained increasing popularity in the fields of Partial Differential Equations (PDEs) and Reduced Order Modeling (ROM), providing domain practitioners with new powerful data-driven techniques such as Neural Operators, Deep Operator Networks (DeepONets) and Deep-Learning based ROMs (DL-ROMs). In this talk, we focus on the latter family of approaches, a collection of techniques built upon the use of deep autoencoders. In particular, starting from their empirical successes, we take a few steps further, addressing questions about the theoretical properties of DL-ROMs and their interaction with the underlying physics of the system. This journey will bring us to a deeper understanding on how to design effective autoencoders (choice of the latent dimension, hyperparameters tuning of convolutional layers, etc.), while also guiding us towards new ways for integrating physical knowledge within DL-ROMs. In particular, we shall describe a novel approach for parametrized Darcy-flow type equations, thanks to which we are able to construct DL-ROMs with exact mass conservation. We conclude by presenting some numerical results related to fluid flows in porous media, ranging from mixed-dimensional problems to nonlinear systems.
