TY - JOUR

T1 - An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning

T2 - Concepts, Implementation and Applications

AU - Samaniego, Esteban

AU - Anitescu, Cosmin

AU - Goswami, Somdatta

AU - Nguyen-Thanh, Vien Minh

AU - Guo, Hongwei

AU - Hamdia, Khader M.

AU - Rabczuk, Timon

AU - Zhuang, Xiaoying

PY - 2020/4/15

Y1 - 2020/4/15

N2 - Partial Differential Equations (PDEs) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behavior of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best-known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate on applications that have an interest for Computational Mechanics. Most contributions explore this possibility have adopted a collocation strategy. In this work, we concentrate on mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. In order to prove the concepts, we deal with several problems and explore the capabilities of the method for applications in engineering.

AB - Partial Differential Equations (PDEs) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behavior of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best-known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate on applications that have an interest for Computational Mechanics. Most contributions explore this possibility have adopted a collocation strategy. In this work, we concentrate on mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. In order to prove the concepts, we deal with several problems and explore the capabilities of the method for applications in engineering.

KW - stat.ML

KW - cs.LG

KW - math.AP

KW - Deep neural networks

KW - Energy approach

KW - Physics informed

UR - http://www.scopus.com/inward/record.url?scp=85077809695&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2019.112790

DO - 10.1016/j.cma.2019.112790

M3 - Article

VL - 362

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 112790

ER -