The principle of minimum potential and complementary energy are the most important variational principles in solid mechanics. The deep energy method (DEM), which has received much attention, is based on the principle of minimum potential energy and lacks the important form of minimum complementary energy. Thus, we propose the deep energy method based on the principle of minimum complementary energy (DCM). The output function of DCM is the stress function that naturally satisfies the equilibrium equation. We extend the proposed DCM algorithm (DCM-P), adding the terms that naturally satisfy the biharmonic equation in the Airy stress function. We combine operator learning with physical equations and propose a deep complementary energy operator method (DCM-O), including branch net, trunk net, basis net, and particular net. DCM-O first combines existing high-fidelity numerical results to train DCM-O through data. Then the complementary energy is used to train the branch net and trunk net in DCM-O. To analyze DCM performance, we present the numerical result of the most common stress functions, the Prandtl and Airy stress function. The proposed method DCM is used to model the representative mechanical problems with the different types of boundary conditions. We compare DCM with the existing PINNs and DEM algorithms. The result shows the advantage of the proposed DCM is suitable for dealing with problems of dominated displacement boundary conditions, which is reflected in theory and our numerical experiments. DCM-P and DCM-O improve the accuracy of DCM and the speed of calculation convergence. DCM is an essential supplementary energy form of the deep energy method. We believe that operator learning based on the energy method can balance data and physical equations well, giving computational mechanics broad research prospects.

中文翻译：

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DCM：基于最小余能原理的深度能量法

最小势能原理和余能原理是固体力学中最重要的变分原理。备受关注的深能法（DEM）是基于最小势能原理，缺少最小余能这一重要形式。因此，我们提出了基于最小互补能量（DCM）原理的深度能量方法。DCM的输出函数是自然满足平衡方程的应力函数。我们扩展了所提出的 DCM 算法 (DCM-P)，在 Airy 应力函数中添加了自然满足双调和方程的项。我们将算子学习与物理方程相结合，提出了一种深度互补能量算子方法（DCM-O），包括分支网络、主干网络、基础网络和特定网络。DCM-O首先结合已有的高保真数值结果，通过数据对DCM-O进行训练。然后利用互补能量在DCM-O中训练支网和主干网。为了分析 DCM 的性能，我们提供了最常见的应力函数、普朗特和艾里应力函数的数值结果。所提出的方法 DCM 用于模拟具有不同类型边界条件的代表性机械问题。我们将 DCM 与现有的 PINN 和 DEM 算法进行比较。结果表明，所提出的 DCM 的优势适用于处理主要位移边界条件的问题，这在理论和我们的数值实验中得到了体现。DCM-P和DCM-O提高了DCM的精度和计算收敛速度。DCM是深层能量法必不可少的补充能量形式。