发表机构
Bauhaus University Weimar; Institute of Structural Mechanics, Bauhaus University Weimar; Fudan University; Institute of Computational Mechanics and Artificial Intelligence, Fudan University; Tsinghua University; Department of Engineering Mechanics, Tsinghua University（魏玛包豪斯大学; 魏玛包豪斯大学结构力学研究所; 复旦大学; 复旦大学计算力学与人工智能研究所; 清华大学; 清华大学工程力学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文系统回顾深度能量法（DEM），将其与PINNs和VPINNs对比，通过增量变分公式扩展至非线性、历史相关及时间相关问题，并以数值示例证明其作为变分问题残差基方法替代方案的有效性。
AI 中文摘要
基于能量的方法为固体和流体力学中出现的、控制方程遵循变分原理的一大类偏微分方程提供了一种自然且物理上一致的框架。与基于残差的物理信息神经网络（PINNs）及其弱形式变体（通过损失最小化来强制执行方程的强形式或弱形式）相比，深度能量法（DEM）直接将解计算为能量或增量势泛函的最小化器。这消除了对残差加权的需求，避免了高阶导数，并通过变分结构直接强制执行物理约束。在本工作中，我们系统地重新审视了深度能量法，将其置于物理信息学习和变分建模的更广泛背景下。我们阐明了DEM、PINNs和VPINNs之间的关系，并确定了能量最小化在稳定性、鲁棒性和可解释性方面具有内在优势的问题类别。特别强调了增量变分公式，这使得DEM能够应用于非线性、历史相关和时间相关问题，包括相场断裂和耗散系统。DEM背后的变分结构进一步为优化和逆问题提供了自然基础，其中能量泛函充当基于物理的约束而非残差惩罚。通过一系列数值示例，我们证明了DEM为变分问题提供了一种有原则且有效的残差基方法替代方案，突出了其相对于现有物理信息方法的优势和局限性。
英文摘要
Energy-based approaches provide a natural and physically consistent framework for a large class of partial differential equations arising in solid and fluid mechanics, where the governing equations follow from variational principles. In contrast to residual-based physics-informed neural networks (PINNs) and their weak-form variants, which enforce the strong or weak form of the equations through loss minimization, the Deep Energy Method (DEM) directly computes the solution as the minimizer of an energy or incremental potential functional. This eliminates the need for residual weighting, avoids high-order derivatives, and enables the direct enforcement of physical constraints through the variational structure.In this work, we systematically revisit the Deep Energy Method, placing it in the broader context of physics-informed learning and variational modeling. We clarify the relationship between DEM, PINNs, and VPINNs, and identify the class of problems for which energy minimization provides intrinsic advantages in terms of stability, robustness and interpretability. Particular emphasis is placed on incremental variational formulations, which allow DEM to be applied to nonlinear, history-dependent and time-dependent problems, including phase-field fracture and dissipative systems. The variational structure underlying DEM further provides a natural foundation for optimization and inverse problems, where the energy functional acts as a physics-based constraint rather than a residual penalty. Through a series of numerical examples, we demonstrate that DEM offers a principled and effective alternative to residual-based methods for variational problems, highlighting its strengths and limitations relative to existing physics-informed approaches.
