Seminário de Avaliação - Série A: Interplay of Physics-Informed Neural Networks and Multiscale Numerical Methods

Orientadores:
Antônio Tadeu Azevedo Gomes - Laboratório Nacional de Computação Científica - LNCC
Frédéric Gerard Christian Valentin - Laboratório Nacional de Computação Científica - LNCC

Banca Examinadora:
Antônio Tadeu Azevedo Gomes - Laboratório Nacional de Computação Científica - LNCC (presidente)
Fabio André Machado Porto - Laboratório Nacional de Computação Científica - LNCC
Alvaro Luiz Gayoso de Azeredo Coutinho - Universidade Federal do Rio de Janeiro - COPPE/UFRJ

Suplentes:
Gilson Antônio Giraldi - Laboratório Nacional de Computação Científica - LNCC

Resumo:

Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general Partial Differentia l Equations (PDEs) by adding them, in some way, as terms from the loss/cost function of a Neural Network (NN). The goal of this work is to explore the PINNs strategy for the resolution of PDE systems whose solutions present challenges for classical numerical methods when applied to physical models containing multiple scales. Typical examples are models with heterogeneous coefficients (oscillatory or high contrast) or singularly perturbed models (boundary layers). Initially, we assess the capacity of vanilla PINNs for approximating these models. We then investigated the reaction-advection-diffusion equation in boundary layers, and by incorporating physical coefficients as predictor variables in a PINN, we are able to obtain good predictions, suggesting potential for parametric studies. However, for highly oscillatory problems, we can observe a substantial impact on the results of PINNs, which fail to adequately approximate problems with multiple scales. Therefore, to mitigate the eff ects of these oscillations, we propose an approach to combine the Multiscale Hybrid-Mixed (MHM) method, which has mathematically proven properties, into conventional PINNs. In summary, within the MHM method, multiscale basis functions on the coarse mesh are obtained by solving completely independent local problems. Here, we propose an approach to estimate these multiscale basis functions through PINN models. Thus, the model is adjusted to generate subsequent basis functions, adapting to the structure and characteristics provided by the local domain, without relying solely on a previous set of training samples. Through numerical validations, we show the ability to approximate the basis functions for Poisson and Helmholtz problems.