This paper concerns the study of history dependent phenomena in heterogeneous materials in a two-scale setting where the material is specified at a fine microscopic scale of heterogeneities that is much smaller than the coarse macroscopic scale of application. We specifically study a polycrystalline medium where each grain is governed by crystal plasticity while the solid is subjected to macroscopic dynamic loads. The theory of homogenization allows us to solve the macroscale problem directly with a constitutive relation that is defined implicitly by the solution of the microscale problem. However, the homogenization leads to a highly complex history dependence at the macroscale, one that can be quite different from that at the microscale. In this paper, we examine the use of machine-learning, and especially deep neural networks, to harness data generated by repeatedly solving the finer scale model to: (i) gain insights into the history dependence and the macroscopic internal variables that govern the overall response; and (ii) to create a computationally efficient surrogate of its solution operator, that can directly be used at the coarser scale with no further modeling. We do so by introducing a recurrent neural operator (RNO), and show that: (i) the architecture and the learned internal variables can provide insight into the physics of the macroscopic problem; and (ii) that the RNO can provide multiscale, specifically FE${}^2$, accuracy at a cost comparable to a conventional empirical constitutive relation.

本文涉及在两尺度设置中研究异质材料中的历史相关现象，其中材料在异质性的精细微观尺度上指定，远小于应用的粗宏观尺度。我们专门研究了一种多晶介质，其中每个晶粒都受晶体塑性控制，而固体则受到宏观动态载荷。均质化理论允许我们直接用本构关系解决宏观问题，本构关系由微观问题。然而，同质化导致了宏观尺度上高度复杂的历史依赖性，这与微观尺度上的历史依赖性可能大不相同。在本文中，我们研究了机器学习的使用，尤其是深度神经网络，利用通过反复求解更精细的比例模型生成的数据来：（i）深入了解控制整体响应的历史依赖性和宏观内部变量；(ii) 创建其解决方案运算符的计算高效代理，无需进一步建模即可直接用于较粗略的规模。我们通过引入递归神经算子 (RNO) 来做到这一点，并表明：(i) 架构和学习到的内部变量可以提供对宏观问题物理的洞察力；(ii) RNO 可以提供多尺度，特别是 FE${}^{\mathrm{2个}}$，精度与传统的经验本构关系相当。