EVENTO
Interplay of Neural Networks and Multiscale Numerical Methods
Tipo de evento: Defesa de Tese de Doutorado
The numerical solution of partial differential equations (PDEs) with multiscale behavior remains a challenge in several scientific areas. Methods based on local decomposition, such as the Multiscale Hybrid-Mixed (MHM) method, offer a robust strategy. In parallel, there is a growing movement toward studying deep learning techniques for solving PDEs. When exploring approaches such as Physics-Informed Neural Networks (PINNs), promising results were observed for reactionadvectiondiffusion equations with boundary layers, especially when physical parameters are incorporated as input variables. However, these models still exhibit limitations for problems with strong oscillations in smallregions. Thus, this thesis proposes a hybrid framework that combines PINNs with the MHM method, approximating multiscale basis functions through models that can adapt to the structure of each local subdomain. Experiments withPoisson and Helmholtz problems demonstrate that PINNs can accurately approximate these basis functions. Even so, the limitations of PINNs prevented a full exploration of the multiscale structures of the method, such as the face partitioning characteristic of MHM. To advance further, a DeepONet-based architecture is proposed, whose ability to learn operators and generalize to different mesh resolutions without retraining proved particularly advantageous when used together with MHM. Results for elliptic and transient problems indicate that the learned operators preserve the expected multiscale behavior and the convergence properties of the original MHM method.
Local: LNCC - Laboratório Nacional de Computação Ciêntifica
Endereço: Getúlio Vargas Av., 333, Quitandinha Petrópolis - Rio de Janeiro CEP 25651-075 - Brasil
Telefone: (24) 2233.6004
Data Início: 28/08/2026 Data Fim: 28/08/2026
Aluno: Larissa Miguez da Silva - - LNCC
Co-Orientador: Frédéric Gerard Christian Valentin - Laboratório Nacional de Computação Científica - LNCC
Orientador: Antônio Tadeu Azevedo Gomes - Laboratório Nacional de Computação Científica - LNCC
Participante Banca Examinadora: Alvaro Luiz Gayoso de Azeredo Coutinho - Universidade Federal do Rio de Janeiro - COPPE/UFRJ Antônio Tadeu Azevedo Gomes - Laboratório Nacional de Computação Científica - LNCC Fabrício Simeoni de Souza. - Universidade de São Paulo - ICMC/USP Gilson Antônio Giraldi - Laboratório Nacional de Computação Científica - LNCC
Suplente Banca Examinadora: Fabio Andre Machado Porto - Laboratório Nacional de Computação Científica - LNCC Maicon Ribeiro Correa - Universidade Estadual de Campinas - UNICAMP
