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Information-Based Symbolic Regression of Local Closures for RANS Transition Modeling

This paper presents an information-based symbolic regression framework that derives compact, interpretable local surrogates for key transition parameters, significantly improving the accuracy of RANS transition modeling for various flow cases while highlighting the need for potential recalibration in low-turbulence scenarios.

Original authors: Javier Capel-Jorquera, José I. Cardesa, M. Chávez-Módena, E. Valero, L. M. González

Published 2026-09-08

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CC BY 4.0

Original authors: Javier Capel-Jorquera, José I. Cardesa, M. Chávez-Módena, E. Valero, L. M. González

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of aerodynamics, the air flowing over a wing or a turbine blade does not always behave the same way. At the start of its journey, the air moves in smooth, orderly layers, a state known as laminar flow. But as it travels, these layers can suddenly break down into a chaotic, swirling mess called turbulence. This shift, known as transition, is a critical moment. It changes how much friction the air exerts on the surface, how heat is transferred, and ultimately, how much lift or drag an object experiences. For engineers designing everything from wind turbines to jet engines, predicting exactly where and when this switch happens is one of the most stubborn challenges in modern physics.

To simulate these flows on a computer, scientists use a set of equations that average out the tiny, rapid fluctuations of turbulence. However, these averaged equations have a blind spot: they cannot naturally see the transition from smooth to chaotic flow. To fix this, engineers add extra rules, called "closures," which act like local sensors. These sensors try to guess the state of the flow using only the information available at a single point, rather than looking at the entire flow field. The problem is that these existing rules are often rough approximations, built on old data and simplified assumptions. They work well enough for some situations but can fail when the conditions change, leading to inaccurate predictions for aircraft or machinery.

A team of researchers from Spain and France has now taken a fresh look at these local sensors. Instead of relying on traditional trial-and-error or complex, unexplainable computer models, they used a method that combines information theory with symbolic regression. Think of this as a way to let the data speak for itself, searching for the simplest, most accurate mathematical sentence that describes the physical reality. By analyzing a vast library of idealized fluid flow solutions, they discovered new, more precise ways to calculate the local state of the air. They then tested these new formulas in a full simulation of a flat plate, a standard test case in aerodynamics. The results showed that while the new formulas were far more accurate at describing the local physics, their success in predicting the actual moment of transition depended heavily on the specific conditions of the flow, revealing that improving one part of a complex system does not always guarantee a better overall result.

The researchers began their work by creating a controlled environment to study the transition. They used a famous family of mathematical solutions known as the Falkner-Skan profiles. These solutions describe how air flows over a flat surface under different pressure conditions, ranging from air being pushed forward to air being slowed down. In this perfect, theoretical world, the researchers knew the exact answer for two key quantities: the momentum thickness, which measures how much the air flow has slowed down due to friction, and a pressure-gradient parameter, which measures how the pressure changes along the surface. These quantities are vital for predicting transition, but in a real computer simulation, they are not directly available. Engineers usually have to estimate them using local clues, like the distance from the wall or the speed of the air nearby.

The team's first step was to determine which local clues contained the most useful information. They treated the problem like a search for the most informative signal. Using a technique based on information theory, they tested thousands of possible combinations of local variables to see which ones best predicted the exact, known answers from their theoretical database. They found that for the pressure-gradient parameter, the most informative clue was a specific combination of how the air speed changed perpendicular to the wall and the square of the distance from the wall. This was not a surprise, as it matched the structure of the existing, older formula, confirming that the old intuition was physically sound. However, for the momentum thickness, the search revealed a better candidate. The traditional method relied on a variable based on the rotation of the fluid, but the new analysis showed that a simple ratio involving the local speed and the distance from the wall was far more consistent and reliable across different flow conditions.

With these superior clues identified, the researchers moved to the second stage: finding the exact mathematical formula that connects the local clues to the true values. They used a tool called symbolic regression, which acts like a digital alchemist. Instead of forcing the data into a pre-chosen shape, this tool generates and tests millions of possible mathematical expressions, keeping only those that are both accurate and simple. The goal was to find a compact equation that could be easily written into computer code. The result was a set of new, explicit formulas that reconstructed the true physical quantities with much higher precision than the standard methods. For the pressure-gradient parameter, the new formula was a complex polynomial that captured subtle nuances the old linear formula missed. For the momentum thickness, the best formula turned out to be a remarkably simple linear relationship, which was both highly accurate and easy to implement.

The true test, however, was not just how well these formulas worked in isolation, but how they performed when inserted into a full, working simulation of a transition model. The team implemented their new formulas into a standard one-equation model used in engineering software and ran simulations of the ERCOFTAC T3 series, a famous set of experiments involving air flowing over a flat plate under various conditions. These experiments included cases with high turbulence, low turbulence, and different pressure gradients. The results were revealing. In several cases, particularly those with moderate to high turbulence or strong pressure gradients, the new formulas improved the prediction of where the transition occurred. The simulation matched the experimental data more closely than the original model did.

Yet, the story was not a simple victory. In two specific cases involving very low turbulence intensity, the new formulas actually made the predictions worse. The transition was predicted to happen too early in one case and too late in the other. This outcome highlighted a crucial lesson in complex modeling: improving a single component does not automatically fix the whole machine. The transition model is a delicate balance of many interacting parts. When the researchers swapped out the local formulas, they changed the balance of the entire system. In some scenarios, the new, more accurate local data exposed weaknesses in other parts of the model, such as how the system estimated the intensity of the incoming turbulence. The researchers noted that the existing rules for estimating this turbulence intensity might need to be updated to match the new, more precise local formulas. Without such adjustments, the benefits of the new formulas could be lost or even reversed.

The study concludes that while information-based symbolic regression is a powerful tool for creating clearer, more accurate local rules, it is not a magic bullet that solves the entire problem of transition modeling. The new formulas provide a more faithful representation of the local physics, offering a solid foundation for future improvements. However, the researchers emphasize that for these new rules to work perfectly in all situations, the rest of the model architecture may need to be recalibrated. The work suggests a path forward where data-driven methods can systematically refine the building blocks of turbulence models, but it also serves as a reminder that in the complex world of fluid dynamics, every piece must fit together perfectly for the whole picture to emerge. The researchers plan to extend their work to more complex, three-dimensional flows and to revisit the other parts of the model to see if they can be improved in the same way, aiming for a future where transition predictions are as reliable and transparent as the physics they describe.

Technical Summary: Information-Based Symbolic Regression of Local Closures for RANS Transition Modeling

Problem Statement
The prediction of laminar–turbulent transition in Reynolds-averaged Navier–Stokes (RANS) simulations remains a critical challenge for aerodynamic design. Local-correlation transition models (LCTMs), such as the one-equation intermittency model, are widely used because they avoid the computational expense of evaluating non-local boundary-layer integrals. However, these models rely on local surrogate variables ( $Re_{\theta L}$ and $\lambda_{\theta L}$ ) to approximate the momentum-thickness Reynolds number ( $Re_\theta$ ) and the Pohlhausen pressure-gradient parameter ( $\lambda_\theta$ ). These surrogates are typically empirical, derived from specific regression problems (e.g., Blasius flow), and may not accurately represent the physics across diverse flow conditions. Furthermore, improving an isolated closure does not guarantee a proportional improvement in the final transition prediction due to the complex coupling with other model components (e.g., onset functions, limiters, and turbulence coupling).

Methodology
The authors propose an information-based symbolic-regression framework to derive compact, interpretable, and more accurate local closures. The workflow consists of three main stages:

Database Generation: A canonical database of Falkner–Skan similarity solutions was generated, covering a wide range of pressure gradients (from near-separation to strongly favorable) and Reynolds numbers. This allows for the direct evaluation of exact integral quantities ( $Re_\theta$ , $\lambda_\theta$ ) and local flow variables.
Information-Theoretic Feature Selection: Using the IT- $\pi$ methodology (based on Rényi mutual information), the authors identified the most informative dimensionless local groups capable of reconstructing the target integral quantities. This step addresses the non-uniqueness of the Buckingham $\pi$ theorem by selecting groups that maximize shared information with the target while enforcing sparsity and physical interpretability.
Symbolic Regression: The identified informative features were used as inputs for PySR (a symbolic regression package) to derive explicit algebraic expressions for the local closures. The resulting candidates were evaluated based on three criteria:
- Reconstruction Accuracy: How well the expression reproduces the target at a design wall-normal location ( $\eta_w^* = 1$ ).
- Activation Coherence: Whether the closure behaves appropriately away from the design location (e.g., not triggering premature transition onset).
- Algebraic Parsimony: Selecting the simplest expression that achieves high accuracy.

Key Contributions and Results

Feature Identification:
- For the pressure-gradient parameter ( $\lambda_\theta$ ), the IT- $\pi$ analysis identified $\Pi^*_\lambda = (\partial v/\partial y) d_w^2 / \nu$ as the dominant informative feature. This recovers the dimensional structure of the baseline closure but confirms its physical relevance through information theory.
- For the momentum-thickness Reynolds number ( $Re_\theta$ ), the analysis identified $Re_{Ue} = U_e d_w / \nu$ as a superior feature compared to the baseline vorticity Reynolds number ( $Re_V = d_w^2 S / \nu$ ). Unlike $Re_V$ , which exhibits a bell-shaped profile that could cause premature activation, $Re_{Ue}$ varies monotonically within the boundary layer, satisfying activation-coherence criteria.
Closure Derivation:
- Pressure Gradient: A high-order symbolic expression was selected for $\lambda_{\theta L}$ , reducing the root-mean-square error (RMSE) by nearly two orders of magnitude compared to the baseline at the design location.
- Momentum Thickness: A simple linear closure, $Re_{\theta L} = Re_{Ue} / 7.710$ , was selected. It achieved an $R^2$ of 0.998 (compared to 0.438 for the baseline) and satisfied activation-coherence constraints.
A Posteriori Assessment (ERCOFTAC T3 Series):
The new closures were implemented in OpenFOAM within the one-equation intermittency model. The results were case-dependent:
- Improvements: The modified model (M15 SR) and a limited recalibrated variant (M15 SR-R, with adjusted $F_{length}$ ) showed improved transition onset predictions for several bypass and adverse-pressure-gradient cases (e.g., T3C2, T3C5).
- Deterioration: Performance degraded for two low-turbulence-intensity cases (T3Am and TSK). The authors attribute this to the fact that under very low free-stream turbulence, other components of the onset mechanism (specifically the turbulence intensity definition $T_u L$ and the critical Reynolds number correlation $Re_{\theta c}$ ) become the limiting factors, masking the benefits of the improved local closures.

Significance and Claims
The paper claims that information-based symbolic regression offers a systematic route to constructing compact, data-informed, and physically transparent local closures. The study demonstrates that:

It is possible to derive local surrogates that are substantially more accurate than baseline empirical expressions when evaluated against canonical similarity solutions.
The "activation coherence" of a closure (its behavior away from the design point) is a critical selection criterion that can be quantified and enforced.
Improving local closures does not automatically improve the global transition prediction; the effect is mediated by the existing model architecture. Consequently, the authors note that while the new closures provide a more interpretable representation of the boundary-layer state, their full potential may require the recalibration of remaining model correlations (such as $Re_{\theta c}$ and $T_u L$ ).

The authors conclude that while the current results are promising, future work must extend the closure-discovery database to non-equilibrium, separated, and three-dimensional flows and address the limitations of the turbulence intensity definition in low-turbulence regimes.

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