Interplay of Physics-Informed Neural Networks and Multiscale Numerical Methods
DOI:
https://doi.org/10.5540/03.2025.011.01.0413Palavras-chave:
Physics-Informed Neural Networks, Multiscale Methods, Surrogate methods.Resumo
Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general Partial Differential Equations (PDEs) by incorporating them as terms in the loss/cost function of a Neural Network (NN). However, they can present some difficulties when the PDEs present multiscale features. To mitigate these issues, we propose an approach to embed PINNs within the framework of the Multiscale Hybrid-Mixed (MHM) method. In the MHM method, multiscale basis functions are obtained on a coarse mesh 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 basis functions, adapting to the structure and characteristics provided by the local domain. Through numerical validations, we show the model’s capability to approximate multiscale basis functions for the Poisson problem.
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