Seminar by Ramon Codina

Speaker

Ramon Codina (Universitat Politècnica de Catalunya)

Title

Stabilization and accuracy enhancement using artificial neural networks for reduced order models in flow problems

Date

  • March 07, 2023 16:00 CET+0100 (Europe/Rome)

  • March 07, 2023 10:00 EST-0500 (US/Eastern)

  • March 07, 2023 09:00 CST-0600 (US/Central)

  • March 07, 2023 07:00 PST-0800 (US/Pacific)

Abstract

Reduced Order Models (ROM) in computational mechanics aim at solving problems approximating the solution in spaces of very low dimension. The idea is to first solve the Full Order Model (FOM) in a high-fidelity space and, from its solution, construct the basis of the ROM space. We shall concentrate on the case in which the FOM is solved by means of a Finite Element (FE) method and the ROM is obtained from a Proper Orthogonal Decomposition (POD) of a series of ’snapshots’, i.e., high-fidelity solutions obtained for example at different time instants or for different values of a parameter of the problem to be solved. This way, the ROM solution can be considered to belong to a subspace of the FOM FE space.

The Variational Multi-scale (VMS) idea is to split the unknown into the resolvable component, in our case living in the FE space, and a remainder, called sub-grid scale (SGS). After setting a problem for the SGS, this problem is what is in fact approximated somehow, so that the SGS can be expressed in terms of the FE solution. When the resulting expression is inserted into the equation projected into the FE space, one ends up with a problem for the FE unknown with enhanced stability problems. The first purpose of this talk is to explain why the VMS strategy can be applied quite naturally to the ROM approximation when this is based in a FE method to approximate flow problems. This yields a stable ROM problem.

The second objective of the talk is to explain how accuracy can be improved using Artificial Neural Networks (ANN). Motivated by the structure of the stabilization terms arising from VMS, an additional correcting term is added to enhance accuracy. This term is an ANN trained with the snapshots, i.e., the high-fidelity solutions used to construct the basis of the ROM. Some numerical examples are presented showing the improvement obtained with the correcting terms.

Recording

Watch the recording on our YouTube channel.