Chiara Giraudo

Stable representations in deep learning

TThis project is placed in the new field of Scientific Machine Learning (SciML) and it aims to investigate different aspects of combining machine learning with numerical methods for partial differential equations. As this discipline is very broad we focus on three different topics:

1. machine learning in MultiGrid or Domain Decomposition solvers, with an emphasis on the training part;
2. combined discretizations by neural network and classical schemes, in particular the Finite Element Method (FEM), with the main focus on training efficiency and stability of the resulting model;
3. machine learning in adaptive FEM.

For the first topic, we start from methods already existing and described in the literature. Our hope is to improve these approaches by overcoming the need for a large amount of data required by using supervised learning in the training of neural networks. One way to accomplish our goal is to use hybrid models. Contrarily to supervised learning, where the neural network is purely trained on data, hybrid models contain some knowledge of the problem, which in theory allows for smaller training sets.

For the second topic (2), we make use of Physics-Informed Neural Networks
(PINNs) and hybrid models to approximate the unknown variables of certain
PDEs employing neural networks. The primary question we are interested in for both the methodologies is the compatibility of discretizations in the sense of classical numerical methods saddle-point problems .

Lastly, we study how machine learning can impact mesh refinement procedures. Our focus verge on r- adaptivity, which redistributes the mesh nodes, and h-adaptivity, which can change the topology of the mesh. Both procedures are strongly dependent on some heuristic or error estimates choices related to the peculiarities of the problem at hand. Therefore, the idea of this project is to study how machine learning can improve performance by substituting such heuristic components. Early recent works in this direction show promising results in the context of h-adaptive methods.