Computer-assisted global regularity across nonlinear families of three-dimensional periodic Navier-Stokes flows

Accurate autoregressive prediction of 3D turbulent flows remains challenging for neural PDE solvers, as small errors in fine-scale structures can accumulate rapidly over rollout. In this paper, we propose FlowRefiner, a flow matching-based iterative refinement framework for 3D turbulent flow simulation. The method replaces stochastic denoising refinement with deterministic ODE-based correction, uses a unified velocity-field regression objective across all refinement stages, and introduces a decoupled sigma schedule that fixes the noise range independently of refinement depth. These design choices yield stable and effective refinement in the small-noise regime. Experiments on large-scale 3D turbulence with rich multi-scale structures show that FlowRefiner achieves state-of-the-art autoregressive prediction accuracy and strong physical consistency. Although developed for turbulent flow simulation, the proposed framework is broadly applicable to iterative refinement problems in scientific modeling.

Cross-Reynolds generalisation in neural PDE solvers remains poorly characterised. On the canonical forced 2D Navier-Stokes benchmark, a trained Fourier Neural Operator reaches 46.68% relative L2 error under a 10x Reynolds-number shift, yet zero-forward-model retrieval baselines already improve to 41-42%. This suggests representation geometry as a major organising variable among the tested methods. We test this hypothesis through ConvAE-Relay, which matches states in a source-trained convolutional autoencoder latent space and borrows dynamics from a source-regime database, achieving 38.34+/-0.07% using only a source-regime database and no target-regime fitting, labels, or database entries. A 2x2 ablation isolates matching quality as dominant over the update rule. Oracle experiments confirm that source-regime dynamics directions remain transferable (cosine similarity ~0.84) when matching stays on-manifold; autoregressive drift is the primary bottleneck (~12 percentage points). From the learned-prediction side, a U-Net with multi-scale skip connections achieves 34.72+/-0.60%, consistent with the retrieval-side finding that local, multi-scale representations organise cross-Reynolds transfer among tested methods. All claims are scoped to this benchmark.

The simulation of fluid flows is computationally expensive due to the complexity of its governing partial differential equations. Machine learning models offer a potential surrogate, enabling learning from simulations and significantly faster predictions of flow fields. However, these models require large training datasets, which introduces a trade-off between dataset generation cost and predictive accuracy. In this work, we investigate the relationship between the size of the training-set and accuracy of the prediction when learning steady flow fields in an industrial-scale stirred vessel. A data set of steady flows is generated using Reynolds Averaged Navier Stokes (RANS) simulations in a range of realistic operating conditions, including impeller speeds and liquid heights. We train implicit neural representations of flow fields and compare purely data-driven and constrained variants. Model performance is evaluated using global mean squared error (MSE), qualitative spatial comparisons of predicted and reference flow fields, and tracer transport simulations. We find that the prediction error decreases monotonically with increasing training data, but also that it exhibits clear diminishing returns beyond moderate dataset sizes. Physics-based constraints significantly improve accuracy and reduce variability across training runs in low-data regimes, and they lead to more stable tracer-transport behavior. Furthermore, reasonable interpolation can be achieved over different impeller speeds and liquid heights. However, these benefits come with an increase in the complexity of training, and their relative advantage diminishes as the training set grows.