Physics-informed neural networks provide a mesh-free framework for solving partial differential equation-governed problems in solid mechanics. However, most existing formulations in linear elasticity still learn the displacement field directly, which does not explicitly exploit the analytic structure of two-dimensional elasticity and becomes restrictive for fracture problems with crack face discontinuities and crack tip singularities. Moreover, existing Kolosov--Muskhelishvili informed neural network formulations still rely on residual-based loss functions with multiple boundary and interface terms, whereas a variational concept has not yet been established. To address these issues, a variational Kolosov--Muskhelishvili informed neural network framework for two-dimensional linear elastic problems with and without cracks is proposed in this work. The solution is represented by two holomorphic Kolosov--Muskhelishvili potentials and trained through an energy-based loss function derived from the principle of minimum total potential energy. For crack problems, a discontinuous stress potential representation is further introduced to embed the crack face condition and crack tip singularity directly into the solution ansatz. The proposed framework is validated on a series of benchmark problems with or without crack problems. The results show that variational Kolosov--Muskhelishvili informed neural network can accurately predict stress and displacement field as well as stress intensity factors. Compared with traditional neural network models, it achieves higher accuracy, simpler loss construction, and faster convergence in the considered cases. Overall, the proposed variational Kolosov--Muskhelishvili informed neural network provides an effective and physically consistent variational framework for two-dimensional linear elastic fracture analysis.

翻译：物理信息神经网络为固体力学中偏微分方程控制问题提供了无网格求解框架。然而，现有线性弹性公式大多仍直接学习位移场，未能显式利用二维弹性的解析结构，且对含裂纹面间断与裂纹尖端奇异性的断裂问题具有局限性。此外，现有Kolosov-Muskhelishvili物理信息神经网络公式仍依赖含多边界与界面项的残差损失函数，尚未建立变分框架。针对上述问题，本文提出了一种面向含/不含裂纹二维线性弹性问题的变分Kolosov-Muskhelishvili物理信息神经网络框架。该框架通过两个全纯Kolosov-Muskhelishvili势函数表示解，并基于最小总势能原理推导的能量损失函数进行训练。针对裂纹问题，进一步引入间断应力势表示，将裂纹面条件与裂纹尖端奇异性直接嵌入解的先验假设中。通过一系列含/不含裂纹的基准问题验证了所提框架的有效性。结果表明，变分Kolosov-Muskhelishvili物理信息神经网络可精确预测应力场、位移场及应力强度因子。与传统神经网络模型相比，在算例中实现了更高精度、更简化的损失构建及更快的收敛速度。总体而言，所提出的变分Kolosov-Muskhelishvili物理信息神经网络为二维线性弹性断裂分析提供了有效且物理一致的变分框架。