Toward a unified data-driven turbulence model through multi-objective learning

TL;DR

The paper introduces a unified, data-driven turbulence model that learns a single, physically consistent closure capable of representing multiple flow mechanisms without manual regioning. It combines a parallel tensor-basis neural network for the Reynolds-stress closure with coupled transport equations, a distribution-based training-set selection over 34 flows, and a regularized ensemble Kalman approach for multi-objective learning from sparse observations, enabling Pareto-optimal performance across regimes. The framework supports additive fine-tuning to create specialist models for targeted flows (e.g., three-dimensional diffusers or external vehicle aerodynamics) while preserving core physical constraints such as the log-law of the wall. Results show robust generalization across attached, separated, secondary, free-shear, and complex 3D flows, with notable improvements over baseline models and demonstrated applicability to NASA Turbulence Modeling Challenger cases; the approach moves turbulence closure toward deployable, broadly applicable tooling, potentially enabling full-device unification with scalable multi-objective optimization. The work also highlights directions for gradient-enhanced training, feature augmentation, and broader objective sets (up to 40), indicating a path to even more comprehensive unification in industrial CFD contexts.

Abstract

Turbulence remains one of the last unresolved problems of classical physics and a major bottleneck to accurate flow prediction in climate, aerospace, and energy systems. Industrial simulations therefore rely on averaged representations of turbulence, which often struggle to predict flows governed by multiple interacting mechanisms. We present a unified, data-driven turbulence modeling framework designed to learn robustly from sparse, indirect observations across diverse flow regimes. The framework embeds physical consistency into a flexible, frame-invariant closure, automatically selects representative training cases based on similarity of flow-feature distributions, and learns a single, unified model through a multi-objective ensemble strategy that balances competing objectives across flows and quantities of interest. The resulting unified foundation model adapts seamlessly across regimes without manual intervention. It outperforms existing turbulence models across a broad spectrum of canonical flows and maintains improved performance in complex three-dimensional configurations of industrial relevance, including a generic car and a gas turbine diffuser. When application-specific accuracy is required, the framework further enables specialist models through additive fine-tuning on targeted flow datasets. The results demonstrate the feasibility of a deployable and generalized turbulence modeling approach that unifies multiple flow mechanisms within a single architecture for a broad range of natural and industrial flows.

Figures (40)

• Figure 1: The proposed framework proceeds through three steps to construct a unified turbulence model. The learning is formulated as a multi-objective optimization problem, yielding a single neural-network-based model that reconciles competing objectives across flows and quantities of interest. (A) First, a physically consistent and frame-invariant model representation is employed, in which the turbulent constitutive relation and transport equations are learned in a coupled and internally consistent manner under physical constraints, enabling a single model to adapt to different flow mechanisms without manual switching. (B) Second, a comprehensive dataset of flows is compiled, and a distance-based training-set selection strategy is used to automatically identify a compact and representative set of training cases by comparing probability distributions of local, frame-invariant flow features, thereby spanning relevant flow physics while avoiding redundancy. (C) Finally, an ensemble-based, multi-objective learning framework is applied to learn a unified model from diverse flows and sparse, indirect observations, balancing competing objectives across flows and quantities of interest. Taken together, these three components yield a unified foundation turbulence model that captures multiple flow regimes within a single set of learned network weights. For application-specific accuracy, the unified foundation model can be adapted into a specialist model through additive fine-tuning (see Fig. S1 in Supplementary Material).
• Figure 2: Overview of the training and evaluation cases and performance of the unified foundation turbulence model. (a) Library of canonical and complex flows used for training and evaluation, comprising 34 cases in total. Canonical flows are grouped into four categories—attached boundary layers, free-shear flows, secondary flows, and separated flows—while two complex three-dimensional flows involving multiple mechanisms are shown in the inner circle. Nine representative cases (highlighted with red circles) form the training set, and the remaining 25 cases are used for testing. (b) Radar chart comparing normalized misfits of the baseline and unified foundation models across flow categories. Smaller radial distances indicate improved performance. The unified foundation model demonstrates robust performance across all categories while maintaining accuracy comparable to the baseline for bump flows, for which only one representative case is shown.
• Figure 3: Performance evaluation of the unified foundation model across representative cases from training flow categories and two complex three-dimensional flows. The model is compared with the ground truth and the baseline model. For attached boundary layers, the flat plate recovers the law of the wall. For free-shear flows, the round jet shows improved streamwise velocity decay along the jet centerline. For separated flows, the periodic hill with a lower slope steepness demonstrates better prediction of separation size and reattachment location. For secondary flows, the square duct at a higher Reynolds number shows accurate recovery of corner vortices. For complex three-dimensional flows, the generic car yields improved separation predictions, while the three-dimensional diffuser matches the ground-truth skin-friction coefficients along the bottom-wall midsection.
• Figure 4: Significant performance improvement of the specialist turbulence model for a complex three-dimensional diffuser flow. Panel A shows the diffuser geometry and the sampling planes used for evaluation, including a vertical spanwise plane near the inclined sidewall (Plane 1) and three streamwise cross-sectional planes within the expansion region (Planes 2--4). Panel B compares flow separation predicted by the baseline model, the specialist model, and the ground truth. The baseline model fails to capture the correct separation behavior, producing spurious sidewall separation and missing the dominant separation pattern. In contrast, the specialist model accurately reproduces both the location and extent of separation observed in the ground truth, correctly capturing the upper-wall separation across all planes. These results demonstrate a large and systematic improvement in predictive accuracy achieved by the specialist model for complex three-dimensional flows.
• Figure 5: Feature-space overlap and conflict resolution enabled by the unified foundation model. Panel A shows a t-SNE projection of the four-dimensional feature space, highlighting overlapping and case-specific regions for the periodic hill (blue) and the square duct (green), with red indicating overlap. Panel B compares model responses in the overlapping feature regions, demonstrating that single-case trained models map similar features to different outputs, which limits generalization, whereas the unified foundation model resolves these conflicts and produces balanced predictions. Panel C shows neuron activation patterns in the sub-network associated with the quadratic coefficient $g^{(2)}$, where periodic hill features activate fewer neurons, while square duct features induce stronger activation and rely more heavily on $g^{(2)}$, reflecting differences in the underlying turbulence mechanisms.