Learning nonlocal constitutive models with neural networks

Xu Hui Zhou, Jiequn Han, Heng Xiao

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


Constitutive and closure models play important roles in computational mechanics and computational physics in general. Classical constitutive models for solid and fluid materials are typically local, algebraic equations or flow rules describing the dependence of stress on the local strain and/or strain-rate. Closure models such as those describing Reynolds stress in turbulent flows and laminar–turbulent transition can involve transport PDEs (partial differential equations). Such models play similar roles to constitutive relation, but they are often more challenging to develop and calibrate as they describe nonlocal mappings and often contain many submodels. Inspired by the structure of the exact solutions to linear transport PDEs, we propose a neural network representing a region-to-point mapping to describe such nonlocal constitutive models. The range of nonlocal dependence and the convolution structure are derived from the formal solution to transport equations. The neural network-based nonlocal constitutive model is trained with data. Numerical experiments demonstrate the predictive capability of the proposed method. Moreover, the proposed network learned the embedded submodel without using data from that level, thanks to its interpretable mathematical structure, which makes it a promising alternative to traditional nonlocal constitutive models.

Original languageEnglish (US)
Article number113927
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Oct 1 2021

All Science Journal Classification (ASJC) codes

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


  • Constitutive modeling
  • Convolutional neural networks
  • Deep learning
  • Inverse modeling
  • Nonlocal closure model


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