This study investigates the use of NeuralUCB for cost-aware large language model (LLM) routing. Existing routing approaches can be broadly grouped into supervised routing methods and partial-feedback methods, each with different tradeoffs in efficiency and adaptivity. We implement a NeuralUCB-based routing policy and evaluate it on RouterBench under a simulated online setting. Experimental results show that the proposed method consistently outperforms random and min-cost baselines in utility reward. Compared with the max-quality reference, our method achieves substantially lower inference cost while maintaining competitive reward. These findings suggest that NeuralUCB is a promising approach for cost-aware LLM routing, while also highlighting remaining challenges in action discrimination and exploration.

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