Learning nonlocal constitutive models with neural networks

TL;DR

This paper introduces a neural network approach to model nonlocal constitutive relations in computational mechanics, capturing complex transport PDE-based behaviors more effectively than traditional models.

Contribution

It proposes a neural network structure inspired by transport PDE solutions to represent nonlocal constitutive models, enabling data-driven learning and interpretability.

Findings

The neural network accurately predicts nonlocal constitutive behaviors.

The model learns embedded submodels without direct data for those levels.

Numerical experiments validate the predictive capability of the approach.

Abstract

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.