PINN-Inspired Topology-Aware Learning for Harmonic State Recognition in Multi-Node Coupled Systems

Accurate dynamic state reconstruction in complex multi-node coupled systems is critical for ensuring operational stability and reliability. However, this task is highly challenging due to spatially sparse measurement sensors, strong dynamic coupling among nodes, and the intractability of explicitly modeling the underlying physical mechanisms. Conventional data-driven methods exhibit limited generalization under sparse labels or out-of-distribution conditions, whereas strict physics-driven solvers often fail to converge in complex environments with unmodeled dynamics. To address these limitations, this paper proposes a physics-informed neural network (PINN)-inspired topology-aware learning framework for multi-node state reconstruction. Rather than acting as a strict physical equation solver, the proposed method innovatively injects physical priors into data-driven temporal modeling. By incorporating physical consistency constraints, latent dynamic regularization, topology-aware priors, and an observer-style multi-branch hybrid fusion strategy, the framework effectively overcomes the drawbacks of single-paradigm models to enhance estimation accuracy and robustness. Extensive experiments on real-world coupled system data demonstrate that the proposed framework outperforms state-of-the-art linear, tree-based, and pure sequential models. Specifically, the proposed topology-aware hybrid observer achieves a Root Mean Square Error (RMSE) ≈ 0.02604 and an R2≈ 0.79060 on the multi-node harmonic reconstruction task, demonstrating superior accuracy and dynamic tracking capability compared to the other baselines. Furthermore, cross-node virtual sensing and ablation experiments verify that the constructed physics-guided observer achieves stable cross-node reconstruction under limited physical observations. The results indicate that integrating PINN-inspired learning with topology-aware modeling provides a highly robust and feasible paradigm for ubiquitous sensing and state estimation in complex networks under restricted measurement conditions.