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.

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