Rotor–gas-bearing systems are governed by a strongly coupled model: a nonlinear Reynolds-type lubrication equation, for the compressible gas-film pressure, and rigid-body equations for the rotor motion, driven by pressure-induced forces. In this setting, the dynamics can be highly force-sensitive: small errors in the pressure field may produce disproportionately large force errors that feed back through the rotor equations, ultimately degrading long-time accuracy and potentially triggering instabilities. As a consequence, standard linear projection-based reduced-order models (pROMs) often require a large number of modes to capture fine pressure structures and to retain stable and faithful long-horizon rotordynamic predictions. In this talk, we propose a nonlinear approximation strategy aimed at reducing the required pROM dimension while preserving stability. We begin with a standard linear pROM built from POD in the offline stage, advanced through residual-minimizing LSPG method in the online stage and combined with ECSW to hyper-reduce the nonlinear residual and Jacobian evaluations. We then move beyond linear trial spaces by introducing a learned nonlinear trial manifold for the pressure field expressed as a graph of a function of a few active POD modes. The resulting manifold pROM is posed as a Galerkin projection to the tangent space of the manifold and solved fully implicitly, as the high-fidelity model, by a quasi-Newton method using manifold-consistent residual, Jacobian and optional curvature information to maintain robustness in strongly nonlinear regimes. Preliminary results on a parameterized rotor--bearing configuration indicate that the nonlinear-manifold pROM retains the accuracy and stability of higher-dimensional linear pROMs with substantially fewer active modes, enabling reliable long-time rotordynamic predictions at significantly reduced online cost.