And-Inverter Graphs (AIGs) are fundamental representations for logic synthesis and verification in Electronic Design Automation (EDA). As structured representations of complex digital systems, AIGs require models to capture functional dependencies beyond local structure and remain robust to functionality-preserving transformations. In learning-based AIG representation, existing approaches are predominantly based on GNNs and rely on local gate-level message passing, limiting their ability to capture circuit-level functional context and making the learned representations sensitive to topology-specific patterns. Therefore, we propose CircuitGate, a function-aware AIG representation learning framework that advances from gate-level semantics to circuit-level functional modeling. CircuitGate explicitly encodes global primary-input (PI) support and models support-overlap-aware reconvergence between fanins, while incorporating logic-inspired Boolean constraints to encourage functionally consistent representations. We evaluate CircuitGate on the large-scale ForgeEDA benchmark and further validate it on the EPFL and ITC'99 benchmarks. Across equivalent-gate identification and signal-probability prediction tasks, CircuitGate consistently outperforms existing methods, achieving up to 21.7% and 14.2% reductions in MAE, respectively. Under direct ForgeEDA-to-OpenABC transfer without fine-tuning, CircuitGate also achieves the best equivalent-gate identification performance, demonstrating strong cross-dataset generalization. These results demonstrate the effectiveness of modeling circuit-level functional dependencies beyond local topology.

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