This paper proposes a method for reconstructing three-dimensional turbulent flows from sparse measurements without the need for ground truth data during training. A weight-sharing network is developed to infer the full flow fields from measurements of velocity sampled at three planes and boundary pressure at one additional plane, inspired by experimental configurations. The weight-sharing network shares identical parameters along homogeneous directions, which results in efficient data utilization and reduced computational memory requirements. First, we compare the weight-sharing network to the PC-DualConvNet, adapted from prior work, by reconstructing a 3D Kolmogorov flow from noise-free measurements with a snapshot-enforced loss. Both networks accurately recover time-averaged 3D flow fields and the correct energy spectrum up to wavenumber 10. The weight-sharing network has the ability to infer flow structures distant from measurement planes. Second, we carry out reconstruction from measurements corrupted with white noise (SNR 15) using a mean-enforced loss. We show that, for the weight-sharing network, validation sensor loss on unseen data decreases with training sensor loss—unlike PC-DualConvNet. This shows that the training sensor loss is a good estimate of the generalization error. The weight-sharing network offers good generalization, parameter efficiency, and hyperparameter robustness. The proposed method opens the possibility of three-dimensional flow reconstruction from experiments.

Simulating turbulent fluid flows is a computationally prohibitive task, as it requires the resolution of fine-scale structures and the capture of complex nonlinear interactions across multiple scales. Consequently, extensive research has focused on analysing turbulent flows from a data-driven perspective. However, due to the complex and chaotic nature of these systems, traditional models often become unstable. To overcome these limitations, we propose a purely stochastic approach that separately addresses the evolution of large-scale coherent structures and the closure of high-fidelity statistical data. To this end, the dynamics of the filtered data are learnt using an autoregressive model. This combines a variational-autoencoder (VAE) and Transformer architecture. The VAE projection is probabilistic, ensuring consistency between the model’s stochasticity and the flow’s statistical properties. The mean realisation of stochastically sampled trajectories from our model shows relative $ {L}_1 $ and $ {L}_2 $ distances of 6% and 10%, respectively. Moreover, our framework enables the construction of meaningful confidence intervals, achieving a prediction interval coverage probability of 80% with minimal interval width. To recover high-fidelity velocity fields from the filtered space, Gaussian Process (GP) regression is employed. This strategy has been tested in the context of a Kolmogorov flow exhibiting chaotic behavior. We compare the performance of our model with state-of-the-art probabilistic baselines, including a VAE and a diffusion model. We demonstrate that our Gaussian process-based closure outperforms these baselines in capturing first and second moment statistics in this particular test bed, providing robust and adaptive confidence intervals.