The three-dimensional modeling of slender structures can be decomposed into a one-dimensional geometrically exact beam problem along the centerline and a cross-sectional variational problem [1]. The cross-sectional state is obtained by minimizing the total potential energy under given centerline parameters, yielding optimal cross-sectional variables consistent with the centerline kinematics. Following the ideas of Basile et al. [2,3], we propose a two-scale computational framework for slender thin-walled beams based on isogeometric analysis (IGA) and physics-informed operator learning. The centerline is described by a discrete elastic rod model, while the cross-section is represented isogeometrically: both the physical domain and displacement fields are expressed as linear combinations of non-uniform rational B-spline (NURBS) basis functions. Consequently, the cross-sectional variational problem reduces to solving for a finite set of NURBS coefficients, converting a high-dimensional field mapping into a finite-dimensional coefficient mapping. This substantially eases operator learning and improves generalization. Unlike purely data-driven methods [4], which rely on labeled data, our approach embeds the potential-energy functional directly into the training objective. This enables unsupervised learning of the operator that maps centerline parameters to optimal cross-sectional variables. The learned operator replaces the repeated solution of cross-sectional boundary-value problems, greatly reducing the cost of two-scale coupling. Through several benchmark problems, we demonstrate the accuracy and efficiency of the proposed method, highlighting nonlinear effects induced by cross-sectional geometry. The framework offers an efficient and robust approach for analyzing slender thin-walled structures.