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Enhancing Symbolic Regression and Universal Physics-Informed Neural Networks with Dimensional Analysis

This paper proposes a framework that integrates dimensional analysis (specifically the Buckingham $\Pi$ theorem and Ipsen's method) with symbolic regression and Universal Physics-Informed Neural Networks to reduce computational costs, minimize overfitting, and significantly improve the accuracy of discovering governing equations from limited or noisy data.

Original authors: Lena Podina, Diba Darooneh, Joshveer Grewal, Mohammad Kohandel

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lena Podina, Diba Darooneh, Joshveer Grewal, Mohammad Kohandel

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 delicious cake, but you only have a list of ingredients and the final taste, not the instructions. You want to write down the exact mathematical formula that turns those ingredients into that taste. This is what scientists call Symbolic Regression: using computers to discover the hidden mathematical equations that explain how the world works.

However, this task is like trying to find a specific needle in a haystack the size of a mountain. The computer has to guess billions of possible formulas, which takes a long time and often leads to wrong answers (overfitting).

This paper introduces a clever trick to make this search easier and more accurate: Dimensional Analysis. Think of this as a "unit translator" that strips away the confusing labels (like meters, seconds, or kilograms) and focuses only on the pure relationships between the numbers.

Here is how the authors did it, using simple analogies:

1. The Problem: Too Many Variables

Imagine you are trying to predict how fast a car will go. You might look at the engine size, the weight of the car, the wind speed, the tire pressure, and the driver's mood. That's five different things to juggle. If you try to find a formula connecting all five, it's a mess.

2. The Solution: The "Buckingham Π" and "Ipsen" Filters

The authors used two specific methods (named after their inventors, Buckingham and Ipsen) to simplify the problem.

The Analogy: Imagine you are baking a cake. Instead of worrying about the exact weight of the flour in grams or the exact temperature in Celsius, you realize that what matters is the ratio of flour to sugar and the ratio of heat to time.
What the paper did: They took their complex data and converted it into these "ratios" (called dimensionless groups). Suddenly, instead of juggling five variables, the computer only had to look at two or three. It's like shrinking that massive haystack down to a small pile of hay.

3. The Experiment: Algebraic Equations (The "Recipe" Test)

First, the team tested this on simple math formulas (like gravity or how a ball falls).

The Result: When they fed the "raw" data (with units) to the computer, it often guessed wrong or took a long time. But when they fed it the "ratio" data (dimensionless), the computer found the exact correct formula almost every time, and it did so much faster. It was like giving the computer a map instead of a blindfold.

4. The Advanced Test: The "Partially Known" Mystery (UPINNs)

Next, they tackled a harder challenge: Differential Equations. These are equations that describe how things change over time, like a population of rabbits growing or a bead spinning on a hoop.

The Scenario: Imagine you know half the recipe (e.g., you know how the rabbit population grows naturally), but you don't know the second half (e.g., how many rabbits are eaten by foxes).
The Tool: They used a special AI called UPINN (Universal Physics-Informed Neural Network). Think of the UPINN as a "smart guesser" that learns the missing part of the recipe based on the data.
The Twist: They applied their "ratio filter" (Dimensional Analysis) before the smart guesser looked at the data.
The Result:
- Without the filter: The smart guesser struggled, especially when the data was noisy (like a recipe with a few burnt crumbs). It couldn't figure out the missing part clearly.
- With the filter: The smart guesser found the missing part much more accurately. Because the data was simplified into pure ratios, the computer didn't get confused by the noise or the units.

5. The Big Win: The "Lotka-Volterra" Example

In their most complex test (a predator-prey model), they showed that by using their method, they could reduce the unknown part of the equation down to having zero extra variables.

The Analogy: Usually, you might think the missing recipe depends on the temperature, the humidity, and the time of day. But their method showed that, actually, the missing part only depends on the ratio of predators to prey. It stripped away all the unnecessary clutter, leaving a crystal-clear, simple formula.

Summary

The paper claims that by translating real-world data into "pure ratios" (dimensionless numbers) before asking a computer to find the math formula, you can:

Find the right answer much more often.
Do it faster because there are fewer possibilities to check.
Handle noisy data better (like a recipe with a few mistakes).
Work even when you only know part of the physics, by helping AI find the missing pieces.

Essentially, they didn't invent a new type of computer; they just taught the computer to look at the problem in a simpler, more "physics-friendly" way, making the job of discovering nature's secrets much easier.

Technical Summary: Enhancing Symbolic Regression and UPINNs with Dimensional Analysis

Problem Statement
In engineering and applied mathematics, deriving accurate, interpretable mathematical models from experimental data is critical for control and optimization. Symbolic regression (SR) has emerged as a powerful tool for discovering closed-form equations directly from data without predefined assumptions. However, SR faces significant challenges: it is an NP-hard problem characterized by an exhaustive search space that leads to high computational costs, slow convergence, and a high risk of overfitting, particularly when datasets are limited or noisy. Furthermore, standard SR algorithms often struggle to enforce physical consistency (dimensional homogeneity) in the discovered equations. While previous works have integrated dimensional analysis into SR, they have largely been limited to fully known equations or have not systematically demonstrated the benefits of nondimensionalization in "partial knowledge" scenarios where some terms of a differential equation are known and others must be learned.

Methodology
The authors propose a framework that integrates dimensional analysis—specifically the Buckingham $\Pi$ theorem and Ipsen's method—with symbolic regression and Universal Physics-Informed Neural Networks (UPINNs). The methodology is divided into two primary workflows:

Full Discovery of Algebraic Equations:
The authors combine the Buckingham $\Pi$ theorem with the PySR symbolic regression algorithm. They demonstrate that transforming raw dimensional data into dimensionless $\pi$ terms prior to regression reduces the number of input variables. This process simplifies the search space, ensuring that candidate expressions are constructed from dimensionless groups rather than original variables, thereby eliminating dimensionally inconsistent expressions automatically.
Partial Knowledge Discovery in Differential Equations (The Main Contribution):
For scenarios where a differential equation is partially known (e.g., $du/dt = P(u,t) + Q(u,t)$ where $P$ is known and $Q$ is unknown), the authors introduce a novel pipeline:
- Nondimensionalization via Ipsen's Method: Unlike the Buckingham $\Pi$ theorem, which minimizes the total number of $\pi$ terms, Ipsen's method allows for the specific selection of dimensionless constants to minimize the number of input variables appearing in the unknown term $Q$ . This is achieved by iteratively eliminating dimensions from the unknown component.
- UPINN Approximation: A Universal Physics-Informed Neural Network is trained to recover the hidden term $Q$ using the nondimensionalized data. The UPINN learns $Q$ as a function of only the relevant dimensionless groups, significantly reducing the complexity of the function approximation task.
- Symbolic Regression: The output of the UPINN (the approximated hidden term) is then fed into a symbolic regression algorithm (PySR) to find a closed-form expression for the dimensionless unknown term.
- Recovery: The final dimensionless equation can be mapped back to the original dimensional variables to recover the governing equation.

The framework was tested on three distinct cases: an extended logistic growth model (first-order ODE), a rotating bead model (second-order ODE), and an extended Lotka-Volterra system (coupled ODEs). Experiments were conducted under both noiseless and noisy conditions.

Key Contributions

Integration of Buckingham $\Pi$ with SR: The paper demonstrates that combining the Buckingham $\Pi$ theorem with PySR significantly improves the accuracy of recovering algebraic equations, often achieving exact recovery where the original formulation fails.
Novel Partial-Knowledge Pipeline: The authors introduce a pipeline integrating Ipsen's method, UPINNs, and SR. This approach specifically targets the recovery of unknown terms in differential equations by reducing the input dimensionality of the unknown component before learning.
Systematic Validation: The study provides a systematic comparison of SR performance on original versus nondimensionalized data across multiple differential equation types, showing consistent improvements in error metrics and equation recovery rates.

Results

Algebraic Equations: When applied to seven algebraic equations, nondimensionalization allowed PySR to recover the true underlying equations exactly (with negligible MSE) in almost all cases, whereas the original formulation frequently failed to generalize or produced incorrect expressions. For example, in the case of Coulomb's law, nondimensionalization reduced the problem to a single $\pi$ term, allowing for exact recovery.
Differential Equations (Logistic Growth): For the extended logistic growth model, the UPINN successfully recovered the hidden term in both original and dimensionless forms. However, symbolic regression on the dimensionless data recovered the true expression ( $\alpha^2/(\alpha^2+1)$ ) with high accuracy, while SR on the original data failed to find the correct form even with increased data volume. This advantage persisted under noisy conditions ( $\epsilon=0.05$ and $\epsilon=0.1$ ), where the dimensionless approach yielded significantly lower MSE.
Differential Equations (Rotating Bead): Similar results were observed for the rotating bead model. The dimensionless approach enabled PySR to recover the true equation ( $-\epsilon \sin \theta + \sin \theta \cos \theta$ ), while the original formulation resulted in high MSE and incorrect expressions.
Differential Equations (Lotka-Volterra): In the extended Lotka-Volterra system, the unknown term was reduced to a parameter-free function of two dimensionless variables ( $x, y$ ). While the UPINN learned the hidden term accurately in both cases, PySR struggled to recover the exact rational function in either formulation, though the dimensionless approach still yielded lower loss values. The authors note that the dataset may have lacked sufficient information to distinguish the ground truth from similar expressions.

Significance and Claims
The paper claims that integrating dimensional analysis with symbolic regression offers a robust framework for the automated discovery of governing equations, particularly when data is limited or noisy. By reducing the search space to dimensionless groups, the method lowers computational costs, mitigates overfitting, and ensures physical interpretability. The authors emphasize that unlike statistical compression methods (e.g., PCA), dimensional analysis preserves the invariant relationships between quantities.

The authors modestly position this work as a proof-of-concept. They acknowledge that the method relies on assumptions such as fully known input variables and dimensional matrices, and that real-world application may require handling hidden or unobserved variables. They also note that while the pipeline improves performance, it does not guarantee the discovery of the true equation in every complex scenario (as seen in the Lotka-Volterra results). Future work is suggested to include relaxing these assumptions, creating benchmarks with real-world data, and investigating the specific effects of different nondimensionalization strategies on algorithm performance.