Identification of parameters in parabolic partial differential equation from final observations using deep learning

Author(s): Khalid Atif1, El-Hassan Essouf2, Khadija Rizki2
1Laboratoire de Mathématiques Appliquées et Informatique (MAI) Université Cadi Ayyad, Marrakech, Morocco
2Laboratoire de Mathématiques Informatique et Sciences de ´lingenieur (MISI) Université Hassan 1, Settat 26000, ´ Morocco
Copyright © Khalid Atif, El-Hassan Essouf, Khadija Rizki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this work, we propose a deep learning approach for identifying parameters (initial condition, a coefficient in the diffusion term and source function) in parabolic partial differential equations (PDEs) from scattered final observations in space and noisy a priori knowledge. In Particular, we approximate the unknown solution and parameters by four deep neural networks trained to satisfy the differential operator, boundary conditions, a priori knowledge and observations. The proposed algorithm is mesh-free, which is key since meshes become infeasible in higher dimensions due to the number of grid points explosion. Instead of forming a mesh, the neural networks are trained on batches of randomly sampled time and space points. This work is devoted to the identification of several parameters of PDEs at the same time. The classical methods require a total a priori knowledge which is not feasible.
While they cannot solve this inverse problem given such partial data, the deep learning method allows them to resolve it using minimal a priori knowledge.

Keywords: deep learning; heat equation; hybrid method; inverse problem; model-driven solution; neural networks; optimization; Tikhonov regularization.