Objective Electrical resistance tomography (ERT) is an advanced sensing technique widely used for monitoring industrial multiphase flows. However, the accuracy of ERT-based flow velocity measurement in pipelines remains limited, primarily due to the inherent ill-posedness of the inverse problem and the simplifying assumptions employed in conventional cross-correlation-based flow velocimetry. These limitations often result in significant estimation errors, particularly under complex flow regimes. To address this challenge, this study proposes a novel flow velocity prediction method that integrates slow feature analysis (SFA) with a convolutional neural network (CNN). The proposed approach aims to overcome the physical constraints of traditional calculation-based methods and significantly improve the accuracy of flow velocity estimation.

Method The proposed SFA−CNN method consists of three main stages: data preprocessing, slow feature extraction, and supervised velocity prediction. First, time-series data are constructed from the differential signals between upstream and downstream ERT measurements. These differential signals capture transient variations induced by the transport of flow structures between two electrode planes, thereby encoding key information related to flow velocity. Second, SFA is applied to extract the slowly varying latent features from the high-dimensional, noise-corrupted ERT measurement sequences. As an unsupervised learning technique, SFA identifies projection directions along which the extracted components exhibit maximal temporal slowness. The resulting slow features represent the intrinsic dynamics of the flow field while suppressing high-frequency noise and irrelevant rapid fluctuations, providing compact, informative, and temporally stable representations of the flow process. Third, the extracted slow features are used as input to a CNN-based regression model for flow velocity prediction. The CNN architecture includes convolutional layers for local feature extraction, pooling layers for dimensionality reduction, and fully connected layers for nonlinear mapping to the target velocity. The integration of SFA and CNN combines the advantages of both unsupervised and supervised learning: SFA provides physically meaningful and noise-robust feature representations, while CNN captures complex spatiotemporal dependencies for accurate regression.

Results To evaluate the performance of the proposed method, extensive simulation experiments were conducted on a solid-liquid two-phase flow experimental platform under various operating conditions, including different flow velocities, phase fractions, and flow regimes. The proposed SFA−CNN method was compared with several benchmark approaches, including a CNN model without SFA preprocessing, the conventional cross-correlation algorithm, and a Transformer-based baseline model. Quantitative evaluation was performed using multiple metrics, including mean relative error, root mean square error (RMSE), and the correlation coefficient between predicted and true velocities. The experimental results demonstrate that the SFA−CNN method consistently achieves superior performance across all test conditions. Specifically, compared with the cross-correlation algorithm, the proposed method reduces the mean relative error by 31.6%, indicating a significant improvement over traditional physical-model-based approach. Compared with a direct CNN model applied to raw ERT measurements, the incorporation of SFA reduces the prediction error by approximately 12.1%, highlighting the effectiveness of unsupervised slow feature extraction in noise suppression and dynamic feature representation. Furthermore, relative to the Transformer-based baseline, which is widely recognized for sequence modeling tasks, the proposed method achieves an 11.4% reduction in mean prediction error, demonstrating its competitiveness and robustness even against state-of-the-art deep learning models. In addition to improved accuracy, the SFA−CNN method exhibits higher prediction stability, with reduced variance in prediction errors across different operating conditions, indicating better generalization capability.

Conclusion The proposed flow velocity prediction method effectively overcomes the physical constraints and computational assumptions inherent in existing calculation-based approaches, such as cross-correlation flow velocimetry, which typically rely on idealized conditions including uniform flow or frozen turbulence. By seamlessly integrating unsupervised slow feature extraction with supervised deep learning, the proposed method expands the applicability of ERT-based velocimetry to more complex and dynamic flow regimes, while significantly improving prediction accuracy. Beyond enhancing ERT-based flow velocity measurement, the SFA-CNN framework provides a general modeling paradigm that combines feature extraction and regression learning in other time-series prediction tasks. Compared with conventional calculation-based methods, the proposed approach holds greater engineering significance and provides a practical and accurate solution for industrial multiphase flow monitoring.