We present the IG-Net, an isogeometric neural network for the analysis of thin shell structures with complex geometries. The proposed approach is built upon spline-based representations and is fully consistent with the isogeometric analysis (IGA), enabling exact geometric descriptions and high-order continuity required for Kirchhoff-Love (KL) shell models. Complex shells are represented using analysis-suitable unstructured T-splines (ASUTS) combined with Bézier extraction, which provides a unified element-wise formulation. Based on these, spline basis functions are explicitly embedded into the network, resulting in a well-defined mathematical interpretation rather than the conventional random neurons. Within the framework, IG-Net supports two computational paradigms. In a physics-informed setting, the network directly solves governing equations of KL shells by minimizing the potential energy. In a data-driven setting, IG-Net can be extended to learn the load-displacement relationship following the deep operator networks (DeepONet) architecture, realizing rapid prediction for the shell deformation under unseen loading scenarios without repeated numerical simulations. Numerical examples involving benchmark and engineering-scale shell structures demonstrate the applicability of the proposed CAD/CAE-integrated framework for the complex shell analysis.