$r-$Adaptive Deep Learning Method for Solving Partial Differential Equations

Deep Learning (DL) [1, 2] is nowadays applied to multiple fields [3], including biomedical applications [4], structural health monitoring [5, 6], and geosteering [7]. Indeed, DL can perform complex tasks with high accuracy without incurring prohibitive computational costs. DL has allowed an essential advance in solving problems where the relationship between input and output data is complex and unknown. For example, merging DL techniques with the Finite Element Method can be used to improve the solution of Partial Differential Equations (PDEs) [8, 9, 10, 11, 12]. In addition, the use of DL to predict PDEs behavior has also raised great interest during the last decade[13, 14, 15]. To solve a PDE using DL, we define a loss function whose global minimum satisfies the PDE and the boundary conditions (BCs). The selection of the numerical method to solve the PDE, formulated in strong, weak, or ultra-weak form, leads to different definitions of the loss function.