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arxiv: 2507.17777 · v2 · submitted 2025-07-22 · 💻 cs.AI

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

Theofanis Aravanis , Grigorios Chrimatopoulos , Mohammad Ferdows , Michalis Xenos , Efstratios Em Tzirtzilakis This is my paper

Pith reviewed 2026-05-19 03:29 UTC · model grok-4.3

classification 💻 cs.AI
keywords symbolic regressionanswer set programminglaminar flowrectangular channelvelocity profilepressure dropfluid mechanicshybrid modeling
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The pith

Symbolic regression combined with answer set programming derives compact equations for laminar channel flow that match analytical solutions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how symbolic regression applied to numerical simulation data for three-dimensional laminar flow in a rectangular channel produces compact expressions for axial velocity and pressure. These expressions recover the expected parabolic velocity profile across the channel and the linear pressure drop along the length, matching known analytical results. To keep the expressions physically valid rather than merely data-fitting, the authors add answer set programming that encodes domain constraints and steers the search. The hybrid method is presented as a way to obtain interpretable, reliable models that respect physical principles in fluid mechanics.

Core claim

Applying symbolic regression to data from numerical simulations of 3D laminar flow in a rectangular channel yields compact symbolic equations for the axial velocity and pressure fields. These equations reproduce the parabolic velocity profile and linear pressure drop with excellent agreement to analytical solutions in the literature. Integrating answer set programming encodes physical constraints so that the generated expressions remain both statistically accurate and physically plausible.

What carries the argument

The SR/ASP hybrid framework, in which symbolic regression generates candidate expressions from a user-defined set of mathematical primitives while answer set programming declaratively enforces domain-specific physical constraints to guide selection toward valid solutions.

If this is right

  • The resulting compact equations allow direct prediction of velocity and pressure without running full numerical simulations each time.
  • Physical constraints enforced by answer set programming reduce the chance of obtaining unphysical or statistically plausible but invalid expressions.
  • The approach supplies interpretable models that can be inspected and potentially extended analytically in fluid mechanics.
  • Similar hybrids could be used on other laminar or steady flow configurations where data are available but governing expressions are sought.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may help discover governing relations in flows where no closed-form analytical solution is known beforehand.
  • Blending generative regression with logical constraint checking could transfer to other physics domains that involve partial differential equations and known invariants.
  • Testing the framework on data from turbulent or geometrically complex flows would reveal whether it can surface previously unrecognized relationships.
  • The combination points toward scientific discovery tools that remain trustworthy by construction rather than by post-hoc validation alone.

Load-bearing premise

That answer set programming can encode the relevant physical constraints for the flow without excluding important but non-obvious relationships or requiring advance knowledge of the exact solution form.

What would settle it

Generate new simulation data for the same rectangular channel at a different aspect ratio or flow rate, then test whether the previously derived symbolic equations still match the new velocity and pressure fields within the reported accuracy.

Figures

Figures reproduced from arXiv: 2507.17777 by Efstratios Em Tzirtzilakis, Grigorios Chrimatopoulos, Michalis Xenos, Mohammad Ferdows, Theofanis Aravanis.

Figure 1
Figure 1. 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). While, as we demonstrate, SR excels in its generative capabilities, it operates within a purely data￾driven manner, potentially overlooking intricate domain-specific constraints and logical relationships inherent to phys… view at source ↗
Figure 2
Figure 2. 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 ex￾pressions that are both accurate and physically valid. clarity; and second, the significant role of knowledge-representation techniques in improving the reli￾ability and domain-specific validity of data-driven SR models. T… view at source ↗
Figure 3
Figure 3. Figure 3: A three-dimensional, symmetrical rectangular duct. Applying the parameters of Equation (3) to Equations (1) transforms the governing equations into their non-dimensional form. Using the chain rule, along with Equation (2), the non-dimensional gov￾erning equations in closed form are obtained, ∂(u 2 ) ∂x + ∂(uv) ∂y + ∂(uw) ∂z = − ∂p ∂x + ∂ 2u ∂x2 + ∂ 2u ∂y2 + ∂ 2u ∂z2 , ∂(vu) ∂x + ∂(v 2 ) ∂y + ∂(vw) ∂z = − ∂… view at source ↗
Figure 4
Figure 4. Figure 4: ). Loss Complexity [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Identity plots for Equations (6) and (7) of the SR models. The red 45◦ line represents a perfect match between the symbolic equation derived from an SR model and the numerical results. Furthermore, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 3D contour-maps of the axial velocity u within the channel, alongside its 2D profile at a rep￾resentative vertical cross-section of the channel, for indicative Reynolds numbers of the testing dataset. Depicted are the predicted quantities as generated by the SR model, and the reference quantities as gen￾erated by the numerical solutions. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: 3D contour-maps of the pressure p within the channel, for indicative Reynolds numbers of the testing dataset. Depicted are the predicted quantities as generated by the SR model, and the reference quantities as generated by the numerical solutions. the robustness of the model and its ability to encode fundamental fluid-dynamic principles in a compact functional form. Taken together, the SR-derived relations… view at source ↗
Figure 8
Figure 8. Figure 8: , ASP operates by defining a logic program, namely, rules and facts that describe the problem domain, allowing an ASP solver to compute stable models, also known as answer sets — i.e., consistent sets of literals (atomic formulae or their negation) that satisfy all given constraints. This capability makes ASP particularly suitable for tasks that require intricate constraint satisfaction, logical inference,… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces a hybrid Symbolic Regression (SR) and Answer Set Programming (ASP) framework to derive compact symbolic expressions for the axial velocity and pressure fields in 3D laminar flow through a rectangular channel, starting from numerical simulation data. It claims these expressions accurately reproduce the expected parabolic velocity profile and linear pressure drop, show excellent agreement with analytical solutions in the literature, and remain physically plausible due to the incorporation of domain-specific constraints via ASP.

Significance. If the quantitative validation and generality of the ASP constraints hold, the work would contribute to interpretable physics discovery in fluid mechanics by showing how declarative constraints can steer SR away from unphysical models. It offers a concrete example of combining data-driven search with knowledge representation for improved reliability in engineering applications. The absence of error metrics and implementation details in the current manuscript limits the immediate impact.

major comments (3)
  1. [Abstract] Abstract: The assertion of 'excellent agreement with analytical solutions' and 'accurately reproducing the expected parabolic velocity profile' is presented without any quantitative error metrics (e.g., L2 norm, maximum relative error, or R² values) for the derived velocity or pressure expressions versus the analytical Poiseuille solution. This omission is load-bearing for the central claim of accuracy and physical plausibility.
  2. [Methods] Methods/Implementation (inferred from workflow description): No details are supplied on the mathematical primitive set for SR, the specific ASP predicates or rules used to encode physical constraints (e.g., no-slip boundaries, incompressibility, or axial momentum balance), or the exact mechanism of hybrid integration. Without these, it is impossible to determine whether the ASP component supplies only general domain constraints or implicitly favors the known quadratic transverse dependence and constant axial gradient of the target solution.
  3. [Results] Results section: The manuscript does not report how the SR/ASP expressions were validated against the simulation data (training vs. test split, cross-validation procedure) or against the analytical solution, nor does it compare performance to pure SR without ASP. This information is required to substantiate that the hybrid approach improves both statistical fit and physical consistency beyond what data-driven SR alone achieves.
minor comments (2)
  1. [Abstract] The abstract uses the phrase 'uncovering hidden physics' while the workflow begins from external numerical data and known analytical targets; a brief clarification of what 'hidden' refers to would improve precision.
  2. [Results] Notation for the channel geometry (e.g., dimensions, coordinate system) and the exact form of the derived symbolic expressions should be stated explicitly in the main text or a table for reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our hybrid SR/ASP framework for discovering physically plausible expressions in laminar channel flow. We address each major comment below and will revise the manuscript to incorporate the requested quantitative metrics, implementation details, and validation comparisons.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The assertion of 'excellent agreement with analytical solutions' and 'accurately reproducing the expected parabolic velocity profile' is presented without any quantitative error metrics (e.g., L2 norm, maximum relative error, or R² values) for the derived velocity or pressure expressions versus the analytical Poiseuille solution. This omission is load-bearing for the central claim of accuracy and physical plausibility.

    Authors: We agree that quantitative error metrics are necessary to rigorously support the claims of accuracy and physical plausibility. In the revised manuscript we will add explicit L2-norm errors, maximum relative errors, and R² values for both the axial velocity and pressure fields when compared to the analytical Poiseuille solution. These metrics will appear in the Results section and will be referenced concisely in the Abstract. revision: yes

  2. Referee: [Methods] Methods/Implementation (inferred from workflow description): No details are supplied on the mathematical primitive set for SR, the specific ASP predicates or rules used to encode physical constraints (e.g., no-slip boundaries, incompressibility, or axial momentum balance), or the exact mechanism of hybrid integration. Without these, it is impossible to determine whether the ASP component supplies only general domain constraints or implicitly favors the known quadratic transverse dependence and constant axial gradient of the target solution.

    Authors: We acknowledge the need for greater transparency in the Methods section. The revised version will explicitly enumerate the mathematical primitives supplied to the symbolic regression engine, list the ASP predicates and rules that encode the no-slip condition, incompressibility, and axial momentum balance, and describe the precise interface by which ASP constraints are injected into the SR search. These additions will demonstrate that the constraints are general physical principles rather than solution-specific biases. revision: yes

  3. Referee: [Results] Results section: The manuscript does not report how the SR/ASP expressions were validated against the simulation data (training vs. test split, cross-validation procedure) or against the analytical solution, nor does it compare performance to pure SR without ASP. This information is required to substantiate that the hybrid approach improves both statistical fit and physical consistency beyond what data-driven SR alone achieves.

    Authors: We agree that a clear validation protocol and baseline comparison are required. The revised Results section will document the data partitioning (training versus held-out test points), any cross-validation procedure employed, and quantitative performance metrics for the hybrid SR/ASP model versus a pure SR baseline without ASP constraints. This comparison will quantify the gains in both statistical accuracy and physical consistency attributable to the ASP component. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation starts from external simulation data

full rationale

The paper applies symbolic regression to numerical simulation data for 3D laminar channel flow, deriving compact expressions for axial velocity and pressure that match expected parabolic and linear profiles. ASP is introduced to enforce declarative physical constraints for plausibility. No load-bearing step reduces by construction to fitted parameters, self-citations, or pre-specified target forms; the workflow remains independent of the final analytical agreement, which is checked against external literature solutions. This is the normal non-circular case for data-driven discovery with added domain constraints.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the domain assumption that numerical simulation data accurately captures the target physics and that ASP can supply effective constraints; no free parameters or new entities are described.

axioms (1)
  • domain assumption Numerical simulation data faithfully represents the 3D laminar flow physics in a rectangular channel.
    Symbolic regression is applied directly to this data to derive the velocity and pressure expressions.

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