A Hybrid Generative Reduced-Order Model for the Minimal Flow Unit
A data-driven reduced-order modelling framework is proposed for wall-bounded turbulent flows to forecast the intermittent near-wall dynamics over extended time horizons from sparse sensor measurements. The approach combines a $β$-VAE-GAN, which compresses high-dimensional flow fields into a low-dimensional latent space, with a sensor-conditioned Transformer that forecasts the evolution of the latent variables. The temporal module employs Easy Attention, a static time-mixing operator that replaces the learnable query-key mechanism of standard self-attention at reduced computational cost, combined with an adapted AdaLN-Zero modulation mechanism for sensor-based conditioning. Evaluated on the Minimal Flow Unit ($Re_τ= 200$) at $y^+ = 14$, the compression stage recovers $87\%$ of the turbulent kinetic energy within a four-dimensional latent space, exceeding the standard $β$-VAE baseline by more than $10\%$. The latent dimensions autonomously encode the characteristic timescales of the flow, with specific coordinates capturing the low-frequency signature of the near-wall regeneration cycle ($T^+ \approx 1724$), establishing the physical interpretability of the learnt representation. The sensor-conditioned Transformer maintains accurate forecasts over $17{,}288\,t^+$ from an initialisation window of only $128\,t^+$, whilst end-to-end inference reconstructs $82\%$ of the turbulent kinetic energy. The principal limitation is the attenuation of rare, extreme-amplitude events, a consequence of the encoder prioritising the most statistically recurrent flow states within the low-dimensional bottleneck. Nevertheless, the framework accurately reproduces the alternating active and quiescent phases of the regeneration cycle, demonstrating its suitability as a surrogate model for the intermittent dynamics of wall-bounded turbulence.