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

This paper introduces a hybrid Symbolic Regression and Answer Set Programming (SR/ASP) framework that successfully uncovers interpretable, physically consistent mathematical laws for 3D laminar fluid flow by combining data-driven formula discovery with declarative reasoning to ensure domain-specific constraints are met.

Original authors: Theofanis Aravanis, Grigorios Chrimatopoulos, Mohammad Ferdows, Michalis Xenos, Efstratios Em Tzirtzilakis

Published 2026-05-05

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Theofanis Aravanis, Grigorios Chrimatopoulos, Mohammad Ferdows, Michalis Xenos, Efstratios Em Tzirtzilakis

Original paper licensed under CC BY 4.0 (http://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

Imagine you are trying to figure out the secret recipe for a perfect cake. You have a giant list of ingredients and the final taste of the cake, but you don't know the instructions.

The Problem: The "Black Box" Mystery
Traditionally, scientists use two main ways to solve this:

The Old Way (Physics): They write down the laws of baking (like "heat makes dough rise") and solve complex math equations. It's accurate, but it's like trying to solve a puzzle while wearing heavy boots—it takes a long time and a lot of effort.
The Modern Way (Machine Learning): They feed the computer thousands of cake examples and let it guess the recipe. It's fast, but the computer acts like a "black box." It gives you the right answer, but it won't tell you why or give you a simple recipe card you can read. It just says, "Trust me, it works."

The New Approach: Symbolic Regression (SR)
This paper introduces a smarter way called Symbolic Regression (SR). Think of SR as a super-smart chef who doesn't just guess the taste but actually writes down the recipe in plain English (or math).

Instead of forcing the computer to fit a specific shape, SR searches through a toolbox of basic math tools (like addition, multiplication, and squaring numbers) to find the simplest, most readable formula that matches the data.

The Goal: To find a short, human-readable equation that explains how the fluid moves, not just where it goes.

The Experiment: The Rectangular Pipe
The researchers tested this on a very specific problem: water flowing smoothly through a rectangular pipe (like a long, flat box).

They used a powerful computer simulation to generate data on how the water moves (speed and pressure) at different speeds.
They fed this data into their SR "chef."
The Result: The SR chef didn't just guess; it wrote down two beautiful, simple formulas.
- One formula described the speed of the water. It looked like a classic "parabola" (a U-shape), which is exactly what physics says should happen in a pipe.
- The other formula described the pressure. It showed a straight, linear drop, which is also exactly what physics predicts.

These formulas were so accurate that they matched the complex computer simulations almost perfectly, but they were simple enough for a human to read and understand instantly.

The Twist: The "Logic Police" (ASP)
Here is the paper's biggest innovation. Sometimes, a computer is so eager to find a pattern that it might invent a "recipe" that works mathematically but makes no sense in the real world. For example, it might suggest that water speed increases if you turn the pipe upside down, which violates the laws of physics.

To fix this, the authors added a second layer called Answer Set Programming (ASP).

The Analogy: Imagine the SR chef writes down 100 possible recipes. The ASP module is like a strict Food Safety Inspector or a Logic Police Officer.
How it works: The Inspector has a rulebook of "Physical Laws" (e.g., "Water must stop at the walls," "Speed must be symmetrical," "Pressure must drop linearly").
The Inspector looks at the 100 recipes the chef wrote and throws out any that break the rules, even if they taste good mathematically.
The Outcome: The final list contains only the recipes that are both accurate (match the data) and logical (obey the laws of physics).

Why This Matters
The paper claims that by combining the "creative guessing" of Symbolic Regression with the "strict rule-following" of Answer Set Programming, they created a system that is:

Fast: It finds the answer quickly.
Accurate: It matches complex simulations.
Honest: It gives you a formula you can actually read and understand, ensuring it doesn't break the laws of nature.

In short, they built a tool that helps scientists discover the "secret recipes" of fluid motion, ensuring the recipes are not just mathematically clever, but physically true.

Technical Summary: ASP-Assisted Symbolic Regression in Fluid Mechanics

Problem Statement
Fluid mechanics modeling faces a dichotomy between traditional Computational Fluid Dynamics (CFD), which is physically transparent but computationally expensive, and modern Machine Learning (ML) approaches, which offer efficiency but often lack interpretability ("black box" behavior). While Symbolic Regression (SR) has emerged as a promising alternative capable of discovering interpretable mathematical expressions from data, purely data-driven SR models risk generating equations that are statistically accurate but physically implausible. They may violate domain-specific constraints or fundamental physical laws because they operate without explicit knowledge of the underlying physics. This study addresses the challenge of modeling three-dimensional (3D) laminar flow in a rectangular channel, specifically focusing on deriving compact, interpretable formulas for axial velocity and pressure fields that are both accurate and physically consistent.

Methodology
The research employs a two-pronged methodology: a standard Symbolic Regression approach followed by an innovative hybrid framework integrating Answer Set Programming (ASP).

Data Generation and Setup:
- The physical problem involves 3D laminar flow in a symmetrical rectangular duct ( $L=5, H=W=1$ ).
- Governing equations are the non-dimensional Navier-Stokes equations and the conservation of mass.
- Numerical solutions are generated using the Finite Volume Method (FVM) implemented in MATLAB with a $50 \times 25 \times 25$ grid.
- The flow is driven by a constant streamwise pressure gradient ( $c$ ), which serves as the primary control parameter. An effective Reynolds number ($Re$) is calculated a posteriori from the numerical solution to serve as a scalar control parameter for the SR models.
- Datasets include training ( $Re \in \{34, 105, 174, 209, 279\}$ ) and testing ( $Re \in \{70, 139, 244\}$ ) sets.
Symbolic Regression (SR) Phase:
- The PySR library (v0.19.4) is used to discover symbolic expressions for axial velocity ( $u$ ) and pressure ( $p$ ) as functions of spatial coordinates ( $X, Y, Z$ ) and $Re$.
- Primitives: The search space is restricted to domain-motivated operators: basic arithmetic ( $+, -, *, /$ ) and low-order polynomials (square, cube).
- Constraints: To ensure interpretability, expression complexity is capped at 25 operations. Nested constraints prevent redundant high-order functions.
- Optimization: Models are evolved over 1000 iterations to minimize Mean Squared Error (MSE), generating a Pareto front of solutions balancing accuracy and simplicity.
Hybrid SR/ASP Phase:
- To address the limitation of purely data-driven models, the authors integrate SR with Answer Set Programming (ASP), a declarative programming paradigm for constraint satisfaction.
- Workflow: SR generates candidate equations. These are parsed into facts (e.g., complexity, loss, presence of specific terms like $Re$ or $Y^2$ ).
- Filtering: An ASP program encodes domain-specific physical constraints (e.g., exclusion of non-physical terms like $X^3$ or $Y^3$ , requirement for $Re$ inclusion, adherence to no-slip boundary conditions, and symmetry requirements).
- Solving: An ASP solver (CLASP) processes these facts and constraints to filter the SR output, retaining only equations that are both statistically accurate and physically valid.

Key Contributions

Derivation of Compact Physical Laws: The study successfully derives explicit, closed-form symbolic equations for 3D laminar flow velocity and pressure that closely match numerical solutions and established analytical trends.
Novel SR/ASP Integration: The paper presents, to the authors' knowledge, the first effort to combine Symbolic Regression with Answer Set Programming. This hybrid framework leverages the generative power of SR and the declarative reasoning of ASP to ensure derived models adhere to physical principles.
Robustness Validation: The approach is validated through noise injection (2% noise added to data) and expansion of the operator set (including trigonometric/exponential functions), demonstrating that the discovered polynomial relationships are genuine reflections of the flow physics rather than artifacts of the search space.

Results

Velocity Model: The SR model derived the equation $u_{symb} = Re(2.18 - 8.46 Y^2)(1 - 3.89 Z^2)$ . This expression captures the expected parabolic velocity profile. It achieved a Normalized Mean Absolute Error (NMAE) of 0.008% on the training set and 0.010% on the testing set.
Pressure Model: The derived pressure equation is $p_{symb} = Re(143.43 - 28.69 X)$ , reflecting a linear pressure drop. This model achieved an NMAE in the range of $10^{-8}\%$ , indicating near-perfect alignment with the numerical data.
Comparison with Literature: The derived velocity equation closely approximates the analytical solution for rectangular ducts found in literature (e.g., $u \propto (1-4y^2)(1-4z^2)$ ), with minor discrepancies attributed to finite-resolution numerical data.
ASP Filtering: In a representative example, the ASP module successfully filtered a Pareto front of 14 equations. It selected the most accurate and physically valid candidates (including the final velocity equation) in approximately 0.1 seconds, a negligible time compared to the 15 minutes required for the SR training phase. The filtering successfully excluded equations violating symmetry or boundary conditions.

Significance and Claims
The paper claims that the proposed SR/ASP methodology demonstrates the potential of combining data-driven and knowledge-representation approaches to enhance the interpretability, reliability, and physical alignment of AI models in fluid dynamics.

Interpretability: Unlike "black box" ML models, the resulting equations are human-readable and reveal the underlying physics (parabolic velocity, linear pressure drop).
Reliability: By integrating ASP, the framework ensures that models do not merely fit the data but also respect domain-specific constraints, preventing the selection of physically impossible equations.
Efficiency: The derived symbolic expressions serve as efficient surrogates for computationally intensive numerical methods, offering rapid and accurate predictions.

The authors maintain a modest tone regarding future implications, noting that while the current study focuses on a specific 3D channel geometry, the framework is generalizable. They suggest future work could involve integrating Chebyshev polynomials into the SR operators and using Large Language Models to automatically derive ASP constraints from natural language descriptions of domain knowledge.

Technical Summary: ASP-Assisted Symbolic Regression in Fluid Mechanics

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