Deep learning-based reconstruction of three-dimensional particle volume fraction fields in solid-liquid two-phase pipe flow from single cross-sectional data

Accurate acquisition of the particle volume fraction (PVF) field of solid–liquid two-phase flow in vertical lift pipes in deep-sea mining is crucial for system transport efficiency and operational safety. However, existing measurements provide only localized information while numerical simulations are computationally intensive, creating the challenge of reconstructing global fields from minimal measurement information. This study developed a deep learning model to reconstruct the global three-dimensional instantaneous Particulate Volume Fraction (PVF) field from single cross-section data. It employs a two-stage architecture, featuring an upsampling module and an enhanced autoencoder with skip connections and multi-head cross-attention, to accurately capture complex particle distributions. The results indicate that the model achieves excellent reconstruction performance with a coefficient of determination R${}^2$ = 0.962 and a mean square error = $3.98\times 10^{-4}$, accurately capturing irregular particle aggregation patterns. Robustness analysis demonstrates stable reconstruction performance across varying input cross-section positions and acceptable accuracy under severe input noise. Furthermore, when the pipe length is extended to 50 times its diameter, a slight performance decrease alongside a linear increase in cost indicates strong potential for application to lengths hundreds of times diameter. Transfer learning evaluation shows efficient adaptation to new particle concentrations using only 10% of training samples, reducing training time by over 90% while maintaining acceptable accuracy.