Physics-informed neural networks (PINNs) provide a flexible framework for solving forward and inverse problems. However, their direct application to structural dynamics remains limited by high system dimensionality and model-form errors arising from incomplete physics. Reduced-order models (ROMs) can alleviate the dimensionality bottleneck, yet existing PINN-ROM couplings typically rely on fixed reduced subspaces, target forward simulations, or assume complete physics, restricting their use for inverse identification under parametric variability or incomplete system knowledge. To address these limitations, this work proposes a Reduced-Order Physics-Informed Neural Network (RO-PINN) framework with adaptive basis refinement for structural identification under known and incomplete physics. Via projection, reduced governing equations are embedded directly into the PINN loss, facilitating learning in a low-dimensional latent space. An adaptive scheme updates the projection basis during training so that the latent space is progressively realigned with evolving structural parameters or learned residual restoring forces. This realignment reduces basis-mismatch errors and limits their influence on the inferred residual force. The method is validated on a four-story steel frame with nonlinear hysteretic braces under sparse and noisy measurements. Results show parameter identification comparable to or more accurate than Bayesian model updating with lower computational cost in the considered cases, recovery of unmodeled nonlinear restoring forces under incomplete physics, and joint identification of residual restoring forces and structural parameters within the same framework. Overall, RO-PINN provides a unified framework for structural identification by integrating reduced-order modeling, adaptive basis refinement, and physics-informed learning within a single formulation.

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