The computation of plasma equilibrium is essential for theoretical research, physical design, and real-time control of magnetic confinement fusion devices. To overcome the significant computational burdens of traditional iteration-based Grad-Shafranov (G-S) solvers and the inherent extrapolation limitations of existing "black-box" data-driven models, this work proposes an explicit, highly efficient, and physics-consistent numerical solver based on the Kolmogorov-Arnold Network (KAN). Using the free-boundary module of Shape Editor (SE), a high-quality dataset comprising 9,292 valid divertor configurations was constructed within a seven-dimensional parameter space via Latin Hypercube Sampling. To ensure physical consistency, a semi-supervised learning framework was developed by embedding G-S equation residuals and plasma current conservation constraints into the loss function. To address the spatial discontinuity of current density at coil boundaries, poloidal field coil currents were superimposed as known source terms rather than being directly predicted. Furthermore, a pruning method utilizing L1 regularization and entropy across all KAN layers was employed to sparsify the network, enabling the transformation of the trained model into explicit analytical expressions via symbolic regression. The symbolic model achieves a high determination coefficient (R²) of 0.9953 for magnetic flux prediction, and error correlation analysis confirms the numerical stability of the symbolic extraction. A comparative analysis with traditional Multilayer Perceptrons further demonstrates the superiority of the KAN architecture in terms of accuracy and interpretability. Based on a linear extrapolation method utilizing first-order gradient continuity, the model's input parameter range is extended beyond the training space. This extrapolated magnetic flux distribution enables the identification of plasma configurations through its positional relationship with the limiter. Validation on 1,000 out-of-distribution test samples yields an overall classification accuracy of 89.1% in distinguishing between divertor and limiter configurations, with a divertor recall rate of 98.9%. A detailed analysis of the misclassified samples reveals the underlying factors affecting extrapolation performance and further delineates the boundary of the model's applicability. Finally, performance benchmarks indicate that the proposed solver reduces single-step computation time from tens of seconds to the millisecond level, with a memory footprint of approximately 10.25 MB. This work provides an efficient and interpretable tool for rapid parameter scans and operational window analysis in tokamaks.