Past Event: Oden Institute Seminar

Deep Learning for Simulation and Inversion Problems

David Pardo, Professor, Mathematics, University of the Basque Country (UPV/EHU), Leioa, Spain Basque Center for Applied Mathematics (BCAM), Bilbao, Spain Ikerbasque (Basque Foundation for Sciences), Bilbao, Spain.

Abstract

Motivation: Inverse problems in which the unknown parameters are connected to experimental measurements through partial differential equations (PDEs) are fundamental for our society. Among others, they serve geophysical engineers to explore the Earth’s subsurface, doctors to examine internal body structures, and civil engineers to detect faults in buildings and structures. All these applications require a real-time interpretation of the experimentally acquired data, as critical decisions depending upon the found parameters have to be made rapidly.

Until recently, inversion methods were based on the repeated solution of forward problems governed by PDEs, and therefore, were computationally expensive. Moreover, inverse problems are generally ill-posed. Nowadays, Artificial Intelligence (AI)-based approaches are the trend in numerical methods for inverse problems governed by PDEs. In particular, deep learning algorithms that incorporate the governing physics laws in the learning process, arose as a promising alternative to traditional methods, as they do not require a mesh, are very fast on GPUs, and are able to approximate complex functions.

Main contribution: Despite these advantages and the promising results obtained for a variety of problems, deep learning for PDEs has strong limitations and challenges. The first part of this presentation will focus on describing some of these challenges, including (a) the control of the quadrature and training errors, (b) the regularity of the approximated solution, and (c) the need for better solvers of nonlinear problems. Essentially, deep learning algorithms lack a solid theoretical background. In addition, their lack of explainability prevents potential users from integrating them into high-risk applications.

The second part of the presentation will show our progress towards overcoming some of the above challenges, giving rise to Deep Learning versions of traditional numerical methods such as: Residual minimization in the dual norm, saddle point optimization, Fourier transforms, Finite Element Methods, Isogeometric Analysis, Ritz Method, Least Squares, and r-adaptivity.

Biography

David Pardo is a Research Professor at Ikerbasque, the University of the Basque Country UPV/EHU, and the Basque Center for Applied Mathematics (BCAM). He has published over 160 research articles and he has given over 260 presentations. In 2011, he was awarded as the best Spanish young researcher in Applied Mathematics by the Spanish Society of Applied Mathematics (SEMA). He leads a European Project on subsurface visualization, several national research projects, as well as research contracts with national and international companies. He is now the PI of the research group on Applied Mathematical Modeling, Statistics, and Optimization (MATHMODE) at UPV/EHU and co-PI of the sister research group at BCAM on Mathematical Design, Modeling, and Simulations (MATHDES).

His research interests include computational electromagnetics, petroleum-engineering applications (borehole simulations), adaptive finite-element and discontinuous Petrov-Galerkin methods, multigrid solvers, deep learning algorithms, and multiphysics and inverse problems.

Links: (a) Ikerbasque Profile, (b) BCAM Profile.