An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications
An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications
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DOI：
10.1016/j.cma.2019.112790
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发表时间：
2020-04-15
影响因子：
7.2
通讯作者：
Rabczuk, T.
中科院分区：
文献类型：
--
作者：
Samaniego, E.;Anitescu, C.;Rabczuk, T.
偏微分方程（PDEs）是对科学和工程中不同现象进行数学建模的基础。解决这些问题是精确了解自然和工程系统行为的关键一步。一般来说，为了将表示真实系统的偏微分方程求解到可接受的程度，分析方法通常是不够的。人们不得不求助于离散化方法。对于工程问题，最有名的选择可能是有限元法（FEM）。然而，强大的替代方法，如无网格方法和等几何分析（IGA）也是可用的。其基本思想是通过专门构建具有某些理想性质的函数来近似求解PDE。在这篇文章中，我们探讨了深度神经网络（dnn）作为近似的一种选择。他们在视觉识别等领域取得了令人印象深刻的成果。dnn在这里被视为函数逼近机。定义它们的结构具有很大的灵活性，并且在架构和实现它们的算法效率方面取得了重要进展，这使得dnn成为近似PDE解的一个非常有趣的替代方案。我们专注于对计算力学感兴趣的应用。大多数探讨这种可能性的文章都采用了并置策略。在这项工作中，我们主要关注力学问题，并分析了PDE的能量形式。机械系统的能量似乎是机器学习方法处理机械问题的自然损失函数。为了证明这些概念，我们处理了几个问题，并探索了该方法在工程应用中的能力。(C) 2019 Elsevier B.V.版权所有
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. (C) 2019 Elsevier B.V. All rights reserved.