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对诸多领域产生了显著影响，特别是在求解固体力学领域的偏微分方程方面，神经网络卓越的逼近能力使其受益匪浅。在求解偏微分方程时，物理信息神经网络和深度能量方法受到了广泛关注。最小势能原理和余能原理是固体力学中两个重要的变分原理。然而，著名的深度能量方法基于最小势能原理，却缺乏最小余能原理这一重要形式。为弥补这一空白，我们提出了基于最小余能原理的深度余能方法。DCEM的输出函数是应力函数，其本身满足平衡方程。我们使用Prandtl和Airy应力函数给出了数值结果，并在建模代表性力学问题时将DCEM与现有的PINNs和DEM算法进行了比较。结果表明，DCEM在应力精度和效率方面优于DEM，并且在处理复杂位移边界条件方面具有优势，理论分析和数值模拟均支持这一结论。我们将DCEM扩展为DCEM-Plus，增加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，我们提出了深度余能算子方法。我们首先使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。