Parametric unsteady flow modeling method for moving-boundary problems based on β-variational autoencoders and transformers

Predicting parameterized flows with moving boundaries requires a compact representation that preserves changes in both geometry and wake dynamics. We develop a nonlinear reduced-order model combining a motion-parameter-conditioned β -variational autoencoder ( β -VAE) with latent-space temporal prediction. A separate branch multiplicatively modulates the encoder mean and standard deviation. The posterior mean defines the state advanced by transformers using easy attention or self-attention and by a long short-term memory (LSTM) network. The framework is evaluated on 2D oscillating-cylinder and flapping-airfoil flows. Three and four latent coordinates, respectively, provide compact vorticity reconstructions. In the predefined extrapolation cases, the mean deterministic reconstruction scores are 91.88% for the cylinder and 90.02% for the airfoil. The corresponding easy attention latent rollout errors are 51.20% and 14.42% over 468 predicted steps following a 32-snapshot initialization. Additional analyses examine parameter conditioning, training-data requirements, window and activation sensitivity, aerodynamic loads, and physical organization of the latent trajectories. The latent geometry retains associations with independently assigned wake regimes and physical observables, while long-horizon errors remain case-dependent. These results support a compact surrogate for the evaluated moving-boundary flows, with explicit limits on extrapolation, load recovery, and recursive stability.