MATERIALS AND METHODS

Physically consistent model representation

For constant-density, incompressible turbulent flows, the mean flow satisfies the Reynolds-averaged Navier–Stokes equations:

(1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} \nabla \cdot \boldsymbol {U} &=& 0,\\ \frac{\mathrm{D}\boldsymbol {U}}{\mathrm{D}t} - \nu \nabla ^2 \boldsymbol {U} + \frac{1}{\rho }\nabla P &=& \nabla \cdot \boldsymbol {\tau }, \end{eqnarray*}\end{document}

with the Reynolds stress \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol {\tau } = -2k!\left(\mathbf {b} + \frac{1}{3}\mathbf {I}\right)$\end{document}. The transport of turbulence scales (the turbulent kinetic energy k and the specific dissipation rate ω) is described by partial differential equations (PDE):

(2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} \boldsymbol {\mathcal {R}}\left[k,\omega ; \, {\boldsymbol{\beta} }(\boldsymbol {q})\right] = \mathbf {0} , \end{eqnarray*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\boldsymbol{\beta}}$\end{document} consists of the constant coefficients in the transport PDE, while the Reynolds stress anisotropy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathbf {b}$\end{document} is modeled using the general eddy-viscosity constitutive relation [22,57]:

(3) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} \mathbf {b} = \sum _{i=1}^{10} g^{(i)}(\boldsymbol {q})\, \mathbf {T}^{(i)} . \end{eqnarray*}\end{document}

The transport equations and constitutive relation are modified simultaneously by formulating the coefficients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\boldsymbol{\beta}}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $g^{(i)}$\end{document} in both relations as function of local features \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol {q}$\end{document} using parallel tensor basis neural networks with essential physical constraints [23,25]. More details can be found in the Section S2.1.

Distribution-based training set selection

Each flow case is represented by the empirical distribution of its local, frame-invariant flow features. At each grid point, an invariant feature vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol {q}(x)\in \mathbb {R}^4$\end{document} is extracted, and the collection of these samples induces an empirical probability measure in feature space. The training case selection procedure proceeds as follows:

• Computing pairwise flow similarity. Similarity between two flow cases with feature distributions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pi _1$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pi _2$\end{document} is quantified using the Wasserstein-2 distance. The squared Wasserstein-2 distance is defined as (4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} W_2^2 = \min _{\gamma \in \Gamma (\pi _1,\pi _2)} \int \Vert \boldsymbol {q}-\boldsymbol {q}^{\star }\Vert ^2 , \mathrm{d}\gamma (\boldsymbol {q},\boldsymbol {q}^{\star }), \end{eqnarray*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol {q}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol {q}^{\star }$\end{document} denote features sampled from these two flow cases, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Gamma (\pi _1,\pi _2)$\end{document} denotes the set of admissible transport plans.
• Clustering and embedding. Pairwise Wasserstein distances between all candidate flow cases are assembled into a symmetric distance matrix. Hierarchical clustering is applied to identify groups of statistically similar flows without prescribing the number of clusters a priori. To aid interpretation of the clustering structure, multi-dimensional scaling is used to embed the flow cases into a 2D plane that approximately preserves the pairwise distribution distances.
• Selecting representative training cases. Representative training cases are selected as cluster medoids, defined as the cases that minimize the sum of distances to all other members within the same cluster.

This procedure ensures coverage of dominant flow mechanisms while avoiding redundant cases and limiting the overall training cost. Details are represented in Section S2.2.

Multi-objective ensemble learning

We couple regularized ensemble learning from indirect data with a multi-objective strategy to balance competing training cases and quantities of interest. Given N objectives, each defined by a training case and a quantity of interest, the update at each iteration proceeds as follows:

• Computing regularized EnKF updates for each objective. For objective j, sparse observations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathbf {y}_j$\end{document} are related to the model parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathbf {w}$\end{document} through an observation operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal {H}_j$\end{document}. An objective-specific ensemble update \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Delta \mathbf {w}_j$\end{document} is computed using the regularized ensemble Kalman filter method, which leverages ensemble covariances and avoids explicit adjoint gradients. The update is obtained by minimizing the cost function with regularization: (5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} \ell _j &=& \big\Vert \mathbf {w}^{i+1}_j - \mathbf {w}^i_j \big\Vert {\mathsf {P}^{-1}}^2\ &&+, \big\Vert \mathbf {y}{! j} - \mathcal {H}j \big[\mathbf {w}^{i+1}{! j} \big] \big\Vert _{\mathsf {R}^{-1}}^2\ &&+, \big\Vert \mathcal {G}\big[\mathbf {w}^{i+1}_j \big] \big\Vert _{\mathsf {Q}^{-1}}^2, \end{eqnarray*}\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal {G}$\end{document} penalizes deviations from a physically meaningful baseline model and stabilizes learning from limited, indirect data.
• Reconciling conflicting objectives by adaptive weighting. Multiple objective-specific updates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lbrace \Delta \mathbf {w}_j\rbrace _{j=1}^N$\end{document} can point in competing directions. To balance them, we seek non-negative weights \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol {\xi } = (\xi _1,\dots ,\xi _N)$\end{document}, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\sum _{j=1}^N \xi _j = 1$\end{document}, such that the combined update direction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\sum _{j=1}^N \xi _j , \Delta \mathbf {w}_j$\end{document} minimizes interference among objectives. Treating each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Delta \mathbf {w}_j$\end{document} as a generalized gradient direction, we normalize the updates to reduce sensitivity to heterogeneous scales and determine \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol {\xi }$\end{document} by minimizing (6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} &&\min _{\boldsymbol {\xi }} \left\Vert \sum _{j=1}^N \xi _j , \Delta \mathbf {w}_j \right\Vert ^2,\ && {\rm s.t.}\ \xi _j \ge 0, \quad \sum _{j=1}^N \xi _j = 1. \end{eqnarray*}\end{document}This auxiliary problem is solved using the Frank–Wolfe algorithm, which iteratively identifies the most conflicting objective and adjusts the weights via an analytic line search, yielding a compromise direction that balances progress across all objectives.
• Forming the unified update. The final model update blends the objective-specific EnKF updates using the learned weights: (7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} \mathbf {w} \leftarrow \mathbf {w} + \eta \sum _{j=1}^{N} \xi _j , \Delta \mathbf {w}_j, \end{eqnarray*}\end{document}where η is a step size. See Section S2.3 for more details.