Carlos Uriarte Baranda

Basque Center for Applied Mathematics, UPV/EHU, Spain

Inverse problems are of great importance to our society. Traditional inverse problem solvers approximate evaluations of the inverse function at certain points. To approximate the full inverse function, which is required in multiple real-time inversion applications, it is possible to use Deep Learning (DL) methods. Critical to the use of DL methods for solving inverse problems is to have a large database. Alternatively, we can use an encoder-decoder-based model. In both cases, we need to

efficiently solve parametric Partial Differential Equations (PDEs), possibly using a DL architecture.

In this work, we propose a DL method for solving parametric PDEs that resembles the Finite Element Method (FEM). The architecture aims to mimic the Finite Element connectivity graph when applying mesh refinements: we associate each Neural Network (NN) layer with mesh

refinement. Each NN layer employs a residual type architecture and extends coarse solutions to finer meshes. For simplicity, we restrict to PDEs with piecewise-constant parameters, which is an important case in multiple applications (e.g., in geophysics). The developed DL-FEM first sets an initial architecture that produces coarse solutions after training. Then, we iteratively and dynamically add layers to the architecture, maintaining the previously trained parameters and adding new ones.

Subsequently, we retrain end-to-end the new model. The training utilizes a combination of customized Adam and SGD optimizers with a loss-dependent adaptive learning rate. We repeat this process until we achieve a desired degree of discretization/accuracy of the parametric solution. Each training step is the equivalent in DL to perform a V-cycle of a multigrid (MG) method.We apply the developed DL-FEM method to a model problem that covers Poisson’s equation, Helmholtz’s equation, and Reaction-Diffusion’s equation. Numerical results show great performance in semi-positive definite (SPD) problems. For non-SPD problems, the method also provides adequate results, and we are currently analyzing their convergence.