ASP-Assisted Symbolic Regression: Uncovering Hidden Physics in Fluid Mechanics

TL;DR

This work applies Symbolic Regression (SR) to uncover compact, interpretable expressions for the axial velocity $u$ and pressure $p$ in laminar 3D channel flow, derived from high-fidelity finite-volume simulations. It demonstrates that SR can yield parabolic cross-section velocity profiles and linear longitudinal pressure drops, closely matching analytical expectations and numerical solutions across multiple Reynolds-number regimes. To mitigate purely data-driven limitations, the authors introduce a hybrid SR/Answer Set Programming (ASP) framework that encodes physical constraints (e.g., No-Slip, symmetry, and laminar scaling) and uses an ASP solver to filter physically plausible SR expressions, achieving rapid, constraint-consistent selection. The results highlight the ability of interpretable, hybrid AI approaches to capture essential fluid-dynamic physics with high accuracy and robustness, offering a scalable path to trustworthy, compact models for complex flows. The methodology holds promise for extending to more complex geometries and could be augmented with adaptive basis functions (e.g., Chebyshev polynomials) and automated constraint derivation via language-model-assisted rule extraction.

Abstract

Symbolic Regression (SR) offers an interpretable alternative to conventional Machine-Learning (ML) approaches, which are often criticized as ``black boxes''. In contrast to standard regression models that require a prescribed functional form, SR constructs expressions from a user-defined set of mathematical primitives, enabling the automated discovery of compact formulas that fit the data and reveal underlying physical relationships. In fluid mechanics, where understanding the underlying physics is as crucial as predictive accuracy, this study applies SR to model three-dimensional (3D) laminar flow in a rectangular channel, focusing on the axial velocity and pressure fields. Compact symbolic equations were derived from numerical simulation data, accurately reproducing the expected parabolic velocity profile and linear pressure drop, and showing excellent agreement with analytical solutions from the literature. To address the limitation that purely data-driven SR models may overlook domain-specific constraints, an innovative hybrid framework that integrates SR with Answer Set Programming (ASP) is also introduced. This integration combines the generative power of SR with the declarative reasoning capabilities of ASP, ensuring that derived equations remain both statistically accurate and physically plausible. The proposed SR/ASP methodology demonstrates the potential of combining data-driven and knowledge-representation approaches to enhance interpretability, reliability, and alignment with physical principles in fluid dynamics and related domains.

Figures (8)

• Figure 1: The SR approach (bottom) can operate, through its derived symbolic expressions, as an efficient, interpretable surrogate for traditional, computationally intensive numerical methods (top).
• Figure 2: Integrated SR and ASP work-flow. SR generates candidate accurate symbolic expressions from fluid-mechanics raw data. Then, ASP applies domain-specific constraints to filter and select expressions that are both accurate and physically valid.
• Figure 3: Geometry of the three-dimensional symmetrical rectangular duct. The computational domain spans $(0,L)\times(-H/2,H/2)\times(-W/2,W/2)$. The duct height $H$ is used as the characteristic length scale.
• Figure 4: The Pareto front (green line) illustrating the optimal trade-off between model loss and complexity. Red markers denote models (equations) on the front, representing the best balance of accuracy and simplicity, while gray points indicate models (equations) that are less optimal in one or both aspects.
• Figure 5: Identity plots for Equations \ref{['eq_u']} and \ref{['eq_p']} of the SR models. The red $45^\circ$ line represents a perfect match between the symbolic equation derived from an SR model and the numerical results.