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.

翻译：最小势能原理与最小余能原理是固体力学中最重要的变分原理。备受关注的深度能量法（DEM）基于最小势能原理，缺乏最小余能这一重要形式。为此，我们提出基于最小余能原理的深度能量法（DCM）。DCM的输出函数为应力函数，可自然满足平衡方程。我们扩展了所提出的DCM算法（DCM-P），在艾里应力函数中增加了自然满足双调和方程的项。将算子学习与物理方程相结合，提出深度余能算子法（DCM-O），包括分支网络、主干网络、基网络和特解网络。DCM-O首先结合现有高保真数值结果通过数据进行训练，随后利用余能训练DCM-O中的分支网络和主干网络。为分析DCM性能，我们给出了最常用应力函数——普朗特应力函数和艾里应力函数的数值结果。所提出的DCM方法被用于模拟具有不同类型边界条件的代表性力学问题。我们将DCM与现有PINNs及DEM算法进行对比，结果表明，DCM在处理位移边界条件主导的问题时具有优势，这一优势在理论与数值实验中均得到体现。DCM-P与DCM-O提升了DCM的精度及计算收敛速度。DCM是深度能量法中余能形式的重要补充。我们相信，基于能量法的算子学习能够良好平衡数据与物理方程，为计算力学提供广阔的研究前景。