A block-coordinate approach of multi-level optimization with an application to physics-informed neural networks

Serge Gratton, Valentin Mercier, Elisa Riccietti () and Philippe L. Toint
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Serge Gratton: Université de Toulouse
Valentin Mercier: Université de Toulouse
Elisa Riccietti: Université de Lyon
Philippe L. Toint: University of Namur

Computational Optimization and Applications, 2024, vol. 89, issue 2, No 4, 385-417

Abstract: Abstract 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 consider two different types of neural architectures, a generic feedforward network and a frequency-aware network. We show that our approach is particularly effective if coupled with these specialized architectures and that this coupling results in better solutions and significant computational savings.

Keywords: Nonlinear optimization; Multi-level methods; Partial differential equations (PDEs); Physics-informed neural networks (PINNs); Deep learning (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-024-00597-1

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