一种用于开发非局部张量本构模型的等变神经算子
An equivariant neural operator for developing nonlocal tensorial constitutive models
AI总结：
提出等变向量云神经网络（VCNN-e）开发非局部张量本构模型，保证平移与置换不变性及旋转等变性，在湍流雷诺应力模拟中验证了其有效性，为RANS方程闭合模型开发铺路。
摘要：
开发鲁棒的本构模型是加速复杂物理模拟的一个基础且长期存在的问题。机器学习为基于各种校准数据构建本构模型提供了有前景的工具。在本研究中，我们提出了一种神经算子，通过具有等变性的向量云神经网络（VCNN-e）来开发张量量的非局部本构模型。VCNN-e尊重本构模型所需的所有不变性属性，忠实地反映了物理上的影响区域，并且适用于不同的空间分辨率。在设计上，该模型保证预测的张量对坐标系的平移和相邻点的排序（置换）是不变的。此外，它对坐标系的旋转是等变的，即输出张量与坐标系共同旋转。我们通过使用VCNN-e模拟湍流的雷诺应力输运模型来评估它，该模型直接计算雷诺应力张量以闭合雷诺平均纳维-斯托克斯（RANS）方程。评估在两种情况下进行：（1）通过使用带闭合模型的雷诺应力输运方程生成的合成数据来模拟雷诺应力模型；（2）通过学习由直接数值模拟生成的数据来预测雷诺应力。对所提出网络的这种先验评估为开发和校准RANS方程的鲁棒、非局部、非平衡闭合模型铺平了道路。
英文摘要：
Developing robust constitutive models is a fundamental and longstanding problem for accelerating the simulation of complicated physics. Machine learning provides promising tools to construct constitutive models based on various calibration data. 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 pave the way for developing and calibrating robust and nonlocal, non-equilibrium closure models for the RANS equations.
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