Regime-Conditioned Variational Operator Learning for Physics-Constrained Prediction and Control of Gas--Liquid Two-Phase Pipe Flow

• Le Hoang Nam

  Thai Nguyen University of Information and Communication Technology, Quyet Thang Ward, Tan Thinh Area, Thai Nguyen, Vietnam

  Author

• Pham Duc Long

  Vinh University, 182 Le Duan Street, Ben Thuy Ward, Vinh, Nghe An, Vietnam

  Author

Two-phase gas--liquid transport in wells, risers, and pipelines exhibits regime transitions that strongly modulate pressure loss, holdup, and control-system observability. Modern sensing provides dense but heterogeneous streams such as differential pressure, temperature, acoustic proxies, and occasional imaging, yet the governing dynamics remain hybrid: continuous conservation laws coupled to discrete, history-dependent flow-pattern switching. This paper develops a regime-conditioned, physics-constrained learning framework that treats the flow regime as a latent stochastic field while preserving conservation and dissipation structure in the continuous states. We formulate a one-dimensional two-fluid backbone with compressibility and inclination, augment it with regime-indexed closure operators for interfacial and wall momentum exchange, and enforce asymptotic and energetic consistency through differentiable inequality constraints. Inference is posed as variational hybrid smoothing, combining a PDE residual penalty with a probabilistic observation model and a spatial--temporal prior over regime transitions. To reduce label dependence, the regime posterior is learned jointly with closure operators via a relaxed discrete representation and an entropy-regularized transition model. The resulting algorithm couples adjoint-based gradients of a stable finite-volume solver to neural operator parameterizations of closures, yielding calibrated uncertainty in both regime assignment and predicted pressure/holdup trajectories. Computational studies on synthetic transients and laboratory-style scenarios indicate improved out-of-sample pressure-gradient prediction and more stable regime probabilities under sparse sensing, with error reductions up to 18\% relative to purely data-driven baselines while maintaining physically admissible dissipation.

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