Static analysis tools have evolved over time to assist in detecting bugs. However, the excessive false warnings can impede developers' productivity and confidence in the tools. Previous research efforts have explored learning-based approaches to identify bug warnings. Nevertheless, their coarse granularity, focusing on either long-term warnings or function-level alerts, is insensitive to individual bugs. Also, they rely on manually crafted features or solely on source code semantics, which is inadequate for effective learning. In this paper, we propose DeepFWI, a learning-based approach that identifies bug-sensitive warnings at a fine-grained granularity. Specifically, we design a novel LSTM-based model that captures multi-modal semantics of source code and warnings from automated static analysis tools (ASATs) and highlights their correlations with cross-attention. To tackle the data scarcity of training and evaluation, we collected a large-scale dataset of 280,273 warnings. We conducted extensive experiments on the dataset to evaluate DeepFWI. The experimental results demonstrate the effectiveness of our approach, with an F1-score 67.06% for confirming true warnings in a finer-grained manner, significantly outperforming all baselines. Additionally, to validate the practicality of DeepFWI from the perspective of developers, we applied DeepFWI to four popular open-source projects. Our approach filtered out the vast majority of warnings, while still successfully surfacing 25 true bug-related warnings that were confirmed through manual analysis.