The variational approach to fracture reformulates Griffith’s criterion into an energy minimization framework. This concept formed the basis of the phase-field fracture (PFF) method, in which cracks are represented by a continuous field ranging from zero (intact state) to one (fully damaged state). Despite its success in predicting complex fracture topologies (e.g., crack merging and branching), conventional PFF models rely on an energy-based formulation that limits their ability to accurately capture strength-driven crack initiation. In addition, they require extremely refined meshes along crack paths, which significantly increase their computational cost. To address these challenges, hybrid neural operator-finite element solvers have recently emerged to accelerate physics-based simulations. In this presentation, we introduce a novel hybrid approach that combines a PFF model based on the Tsai-Wu failure criterion with a deep neural operator surrogate to efficiently capture crack nucleation in orthotropic materials. Orthotropic mechanical behavior is incorporated through direction-dependent stiffness, material strength, and a physics-based critical energy-release rate. The approach is evaluated using benchmark problems to study the impact of directional material properties on fracture behavior. A deep neural operator surrogate is employed to efficiently solve the governing phase-field equations, demonstrating accurate and computationally efficient predictions of fracture initiation and propagation across different material orientations.