We present a new approach for predictive modeling and its uncertainty quantification for mechanical systems, where coarse-grained models such as constitutive relations are derived directly from observation data. We explore the use of a neural network to represent the unknown constitutive relations, compare the neural networks with piecewise linear functions, radial basis functions, and radial basis function networks, and show that the neural network outperforms the others in certain cases. We analyze the approximation error of the neural networks using a scaling argument. The training and predicting processes in our framework combine the finite element method, automatic differentiation, and neural networks (or other function approximators). Our framework also allows uncertainty quantification in the form of confidence intervals. Numerical examples on a multiscale fiber-reinforced plate problem and a nonlinear rubbery membrane problem from solid mechanics demonstrate the effectiveness of our framework.

Daniel Z. Huang

Kailai Xu

Charbel Farhat

Eric Darve

Journal of Computational Physics

JOURNAL OF COMPUTATIONAL PHYSICS 彻底处理了物理问题的计算方面，介绍了物理学各个领域中出现的数学方程的数值求解技术。该杂志力求强调跨越学科界限的方法。《计算物理学杂志》还出版 4 页或更少的简短注释（包括图表、表格和参考文献，但不包括标题页）。对已在本期刊上发表的文章发表评论的致编辑的信件也将被考虑。笔记和信件都不应该有摘要。

The Journal of Computational Physics focuses on the computational aspects of physical problems, presenting numerical solution techniques for mathematical equations arising in various fields of physics. The journal emphasizes interdisciplinary approaches. Additionally, the Journal of Computational Physics accepts brief notes of up to 4 pages (including figures, tables, and references, but excluding the title page). Letters to the editor commenting on articles published in this journal will also be considered. Notes and letters should not include abstracts.

《Journal of Computational Physics》专注于物理问题的计算方法，介绍了在物理学各个领域中出现的数学方程的数值求解技术。该杂志强调跨学科的方法。此外，《Journal of Computational Physics》还接受最多4页的简短注释（包括图表、表格和参考文献，但不包括标题页）。对已在本期刊上发表的文章的评论信件也将被考虑。注释和信件不需要摘要。

Category	Quartile
物理与天体物理	2
物理与天体物理, 物理数学物理	1
物理与天体物理, 计算机跨学科应用	2

Learning constitutive relations from indirect observations using deep neural networks

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Journal Info

Category	Quartile
PHYSICS, MATHEMATICAL	1