A Block-Coordinate Approach of Multi-level Optimization with an Application to Physics-Informed Neural Networks

Serge Gratton (1, 2) , Valentin Mercier (2, 3) , Elisa Riccietti (4) , Philippe L Toint (5)

1 IRIT-APO - Algorithmes Parallèles et Optimisation
2 Toulouse INP - Institut National Polytechnique (Toulouse)
3 CERFACS - Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique
4 OCKHAM - Optimisation, Connaissances pHysiques, Algorithmes et Modèles
5 NaXys - Namur Center for Complex Systems [Namur]

Multi-level methods are widely used for the solution of large-scale problems, because of their computational advantages and exploitation of the complementarity between the involved sub-problems. After a re-interpretation of multi-level methods from a block-coordinate point of view, we propose a multi-level algorithm for the solution of nonlinear optimization problems and analyze its evaluation complexity. We apply it to the solution of partial differential equations using physics-informed neural networks (PINNs) and show on a few test problems that the approach results in better solutions and significant computational savings.

Mots clés: partial differential equations; nonlinear optimization; multi-level methods; physics-informed neural networks (PINNs); deep learning

Domaines: Informatique [cs]; Mathématiques [math]

Serge Gratton, Valentin Mercier, Elisa Riccietti, Philippe L Toint. A Block-Coordinate Approach of Multi-level Optimization with an Application to Physics-Informed Neural Networks. 2023. ⟨hal-04104450⟩