In recent years, the rapid advancement of deep learning has significantly impacted various fields, particularly in solving partial differential equations (PDEs) in the realm of solid mechanics, benefiting greatly from the remarkable approximation capabilities of neural networks. In solving PDEs, Physics-Informed Neural Networks (PINNs) and the Deep Energy Method (DEM) have garnered substantial attention. The principle of minimum potential energy and complementary energy are two important variational principles in solid mechanics. However, the well-known Deep Energy Method (DEM) is based on the principle of minimum potential energy, but there lacks the important form of minimum complementary energy. To bridge this gap, we propose the deep complementary energy method (DCEM) based on the principle of minimum complementary energy. The output function of DCEM is the stress function, which inherently satisfies the equilibrium equation. We present numerical results using the Prandtl and Airy stress functions, and compare DCEM with existing PINNs and DEM algorithms when modeling representative mechanical problems. The results demonstrate that DCEM outperforms DEM in terms of stress accuracy and efficiency and has an advantage in dealing with complex displacement boundary conditions, which is supported by theoretical analyses and numerical simulations. We extend DCEM to DCEM-Plus (DCEM-P), adding terms that satisfy partial differential equations. Furthermore, we propose a deep complementary energy operator method (DCEM-O) by combining operator learning with physical equations. Initially, we train DCEM-O using high-fidelity numerical results and then incorporate complementary energy. DCEM-P and DCEM-O further enhance the accuracy and efficiency of DCEM.

翻译：近年来，深度学习的快速发展显著影响了多个领域，尤其在固体力学偏微分方程求解中，得益于神经网络的卓越逼近能力。在求解偏微分方程时，物理信息神经网络（PINNs）和深度能量法（DEM）获得了广泛关注。最小势能原理和最小余能原理是固体力学的两个重要变分原理。然而，著名的深度能量法基于最小势能原理，但缺乏最小余能这一重要形式。为弥补这一空白，我们提出基于最小余能原理的深度余能法（DCEM）。DCEM的输出函数为应力函数，其自然满足平衡方程。我们利用普朗特应力函数和艾里应力函数给出数值结果，并将DCEM与现有的PINNs和DEM算法在典型力学问题建模中进行比较。结果表明，DCEM在应力精度和效率方面优于DEM，且在复杂位移边界条件处理上具有优势，这得到了理论分析和数值模拟的支持。我们将DCEM扩展为DCEM-Plus（DCEM-P），新增了满足偏微分方程的项。此外，通过将算子学习与物理方程相结合，我们提出深度余能算子方法（DCEM-O）。首先利用高保真数值结果训练DCEM-O，然后引入余能。DCEM-P和DCEM-O进一步提升了DCEM的精度和效率。