<p>During the installation of offshore wind turbine blades in the hammerhead configuration, wave-induced tower-top motions can exhibit strongly coupled fore-aft and side-side dynamics that manifest as complex orbital trajectories. These motions constrain operational weather windows, increase installation risk, and pose challenges for real-time decision-making. Consequently, accurate and computationally efficient prediction of tower-top response is critical during installation operations. This paper presents a physics-informed autoencoder framework as a foundational component for digital twin development of tower-top motions during offshore wind turbine installation. A large-scale numerical simulation campaign is conducted to generate tower-top displacement time series across a broad range of sea states, wave directions, and stochastic wave realizations. Reduced-order governing equations describing the coupled horizontal motion are identified directly from simulation data using sparse system identification and parameterized as smooth functions of key sea-state variables. These data-derived dynamics are embedded as physics-based constraints within the autoencoder training process, guiding the latent representation toward physically consistent behavior. The resulting model learns compact latent representations of coupled tower-top motion that preserve dominant temporal and spectral characteristics while enabling efficient computation. Across the evaluated conditions, the framework achieves millimeter-level reconstruction accuracy and demonstrates robust consistency with respect to varying sea states and stochastic wave realizations. By combining reduced-order physics with representation learning, the proposed approach establishes a practical foundation for installation-phase digital twins, enabling fast and physically consistent prediction of tower-top motions under evolving environmental conditions.</p>