Real-time control of multiphase processes with learned operators

Abstract

Multiphase flows frequently occur naturally and in manufactured devices. Controlling such phenomena is extremely challenging due to the strongly non-linear dynamics, rapid phase transitions, and the limited spatial and temporal resolution of available sensors, which can lead to significant inaccuracies in predicting and managing these flows. In most cases, numerical models are the only way to access high spatial and temporal resolution data to an extent that allows for fine control. While embedding numerical models in control algorithms could enable fine control of multiphase processes, the significant computational burden currently limits their practical application. This work proposes a surrogate-assisted model predictive control (MPC) framework for regulating multiphase processes using learned operators. A Fourier Neural Operator (FNO) is trained to forecast the spatiotemporal evolution of a phase-indicator field (the volume fraction) over a finite horizon from a short history of recent states and a candidate actuation signal. The neural operator surrogate is then iteratively called during the optimisation process to identify the optimal control variable. To illustrate the approach, we solve an optimal control problem (OCP) on a two-phase Eulerian bubble column. Here, the controller tracks piecewise-constant liquid level setpoints by adjusting the gas flow rate introduced into the system. The results we obtained indicate that field-level forecasting with FNOs are well suited for closed-loop optimization since they have relatively low evaluation cost. The latter provide a practical route toward MPC for fast multiphase unit operations and a foundation for future extensions to partial observability and physics-informed operator learning.

Figures (7)

• Figure 1: Representative snapshots of the bubble-column phase field (volume fraction) under closed-loop operation, shown for different commanded liquid levels. The interface location and the internal recirculation/bubble-plume structures evolve nonlinearly with the inlet actuation, highlighting the strongly state-dependent hydrodynamics captured by the CFD dataset.
• Figure 2: Validation mean squared error (MSE) as a function of training epoch for different learning rates. All configurations show a rapid initial drop in error followed by gradual convergence toward a similar low-MSE plateau. Larger learning rates accelerate early convergence, while smaller learning rates reduce the error more gradually; overall, learning rates in the intermediate range provide the best balance between convergence speed and final validation performance.
• Figure 3: Comparison of overall prediction accuracy on the validation and test sets using mean squared error (MSE), mean absolute error (MAE), and structural similarity index (SSIM). The similar values across both splits suggest stable performance and good generalization of the forecasting model.
• Figure 4: Volume fraction field forecasting over a 5-step horizon with the trained Fourier Neural Operator (FNO). Top row: ground-truth volume fraction fields at $t\!+\!1$ to $t\!+\!5$ from the CFD model. Bottom row: corresponding FNO predictions, demonstrating the surrogate’s ability to propagate interface motion and large-scale structures.
• Figure 5: Closed-loop setpoint tracking performance for the bubble-column liquid level. The measured level (blue) is regulated to a piecewise-constant reference (red dashed) across repeated setpoint changes, with short transients at switching times and small steady-state offsets near the operating extremes.