An equivariant neural operator for developing nonlocal tensorial constitutive models

Abstract

Developing robust constitutive models is a fundamental and longstanding problem for accelerating the simulation of multiscale physics. Machine learning provides promising tools to construct constitutive models based on various calibration data. However, with either traditional or machine learning techniques, the constitutive model for tensorial quantities has been much less studied than scalar quantities due to more complicated physics, even though the former plays an important role in many scientific and engineering applications. In this work, we propose a neural operator to develop nonlocal constitutive models for tensorial quantities through a vector-cloud neural network with equivariance (VCNN-e). The VCNN-e respects all the invariance properties desired by constitutive models, faithfully reflects the region of influence in physics, and is applicable to different spatial resolutions. By design, the model guarantees that the predicted tensor is invariant to the frame translation and ordering (permutation) of the neighboring points. Furthermore, it is equivariant to the frame rotation, i.e., the output tensor co-rotates with the coordinate frame. We evaluate the VCNN-e by using it to emulate the Reynolds stress transport model for turbulent flows, which directly computes the Reynolds stress tensor to close the Reynolds-averaged Navier–Stokes (RANS) equations. The evaluation is performed in two situations: (1) emulating the Reynolds stress model through synthetic data generated from the Reynolds stress transport equations with closure models, and (2) predicting the Reynolds stress by learning from data generated from direct numerical simulations. Such a priori evaluations of the proposed network on realistic and challenging datasets pave the way for developing and calibrating robust and nonlocal, non-equilibrium constitutive models for the RANS equations and other mechanical problems.