Arctic rivers represent important components of the Arctic and global hydrological and climate systems, serving as dynamic conduits between terrestrial and marine environments in some rapidly changing regions. They transport freshwater, sediments, nutrients, and carbon from vast watersheds to the Arctic Ocean and affect ocean circulation patterns and regional climate dynamics. Despite their importance, modeling Arctic rivers remains challenging because of sparse data networks, unique cryospheric dynamics, and complex responses to hydrometeorological variables. In this study, a novel hybrid deep learning model is developed to address these challenges and predict Arctic River discharge by incorporating Kolmogorov-Arnold Networks (KAN), Long Short-Term Memory, and the attention mechanism with seasonal trigonometry encoding and physics-based constrains. It integrates several novel components: 1) The KAN-based deep learning component learns and captures intricate temporal patterns from nonlinear hydrometeorological data; 2) Explicit physical constrains designed for the characteristics of permafrost-dominated watersheds govern snow accumulation and melt processes through the architectural design and loss function; 3) The seasonal variations are accounted for using trigonometry functions to represent cyclical patterns; 4) A residual compensation stricture allows the proposed model to revisit systematic errors in initial predictions and helps capture complex nonlinear processes that are not fully represented. The Kolyma River, which is significantly dominated by the permafrost, is adopted to test the performance of the newly developed model. It obtains more robust and accurate predictive performance compared to baseline models. The role of physical constraints, the residual compensated architecture, and the trigonometry encoding are assessed by ablation analysis. The results indicate that these components positively contribute to improving the predictive performance. This novel approach addresses the unique challenges of hydrological forecasting in cold, permafrost-dominated regions and provides a robust framework for predicting Arctic River discharge under changing climate conditions.