Physics-constrained deep learning for prediction of three-dimensional homogeneous isotropic turbulence

Abstract

Three-dimensional turbulence is inherently chaotic and multiscale, posing significant computational challenges for traditional simulation methods, such as direct numerical simulation (DNS), large eddy simulation, and Reynolds-averaged Navier–Stokes models. Although deep learning (DL) offers a promising data-driven alternative, most existing studies have focused on two-dimensional flows, limiting the physical fidelity and applicability to realistic turbulent processes. This article presents a physically constrained DL framework for predicting three-dimensional homogeneous isotropic turbulence. A neural network was augmented with a Fourier projection to enforce incompressibility and was trained using a two-term loss function that accounted for both the velocity and vorticity fields. The model was trained on DNS data at Reλ = 58, and its generalizability was evaluated on higher-Reynolds-number datasets at Reλ = 92, 160, and 433, where the larger domain was divided into subvolumes matching the training size, the flow fields of which were predicted separately, and then concatenated to reconstruct the full domain. Temporal sampling over several finite lead times enabled the forecasting capability of the model to be evaluated. The proposed approach accurately reproduces key turbulence statistics, resolves three-dimensional vortex structures, and maintains divergence-free velocity fields across prediction horizons up to a lead time of 0.3 Eulerian integral timescale. The results demonstrate that combining data-driven DL with physics-based constraints enables the efficient and reliable prediction of three-dimensional turbulence, balancing computational efficiency and physical fidelity.