Multiscale problems are widely observed across diverse domains in physics and engineering. Translating these problems into numerical simulations and solving them using numerical schemes, e.g. the finite element method, is costly due to the demand of solving initial boundary-value problems at multiple scales. On the other hand, multiscale finite element computations are commended for their ability to integrate micro-structural properties into macroscopic computational analyses using homogenization techniques. Recently, neural operator-based surrogate models have shown trustworthy performance for solving a wide range of partial differential equations. In this work, we propose a hybrid method in which we utilize deep operator networks for surrogate modeling of the microscale physics. This allows us to embed the constitutive relations of the microscale into the model architecture and to predict microscale strains and stresses based on the prescribed macroscale strain inputs. Furthermore, numerical homogenization is carried out to obtain the macroscale quantities of interest. We apply the proposed approach to quasi-static problems of solid mechanics. The results demonstrate that our constitutive relations-aware DeepONet can yield accurate solutions even when being confronted with a restricted dataset during model development.

翻译：多尺度问题在物理学和工程学的多个领域中广泛存在。由于需要在多个尺度上求解初边值问题，将这些问题转化为数值模拟并使用数值格式（如有限元法）进行求解的成本高昂。另一方面，多尺度有限元计算因其能够通过均匀化技术将微观结构特性整合到宏观计算分析中而受到推崇。近年来，基于神经算子的代理模型在求解各类偏微分方程方面已展现出可靠的性能。在本工作中，我们提出了一种混合方法，利用深度算子网络对微观尺度物理进行代理建模。这使得我们能够将微观尺度的本构关系嵌入到模型架构中，并根据给定的宏观尺度应变输入来预测微观尺度的应变和应力。此外，通过执行数值均匀化来获得感兴趣的宏观尺度物理量。我们将所提出的方法应用于固体力学的准静态问题。结果表明，即使在模型开发过程中面临有限的数据集，我们提出的本构关系感知DeepONet仍能产生精确的解。