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

arXiv: https://arxiv.org/abs/2411.15919 · HTML: https://mlsanity.com/paper/2411.15919/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 42 references that are in the map.
- First posted on arXiv 24 Nov 2024.

## Builds on (most cited first, 14 of 14)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [SymbolicGPT: A Generative Transformer Model for Symbolic Regression](https://mlsanity.com/paper/2106.14131/) (2021)
- [On Interpretability of Artificial Neural Networks: A Survey](https://mlsanity.com/paper/2001.02522/) (2020)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws](https://mlsanity.com/paper/2303.03192/) (2023)
- [A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge](https://mlsanity.com/paper/2209.06257/) (2022)
- [Dimensionally Consistent Learning with Buckingham Pi](https://mlsanity.com/paper/2202.04643/) (2022)
- [A Neural-Guided Dynamic Symbolic Network for Exploring Mathematical Expressions from Data](https://mlsanity.com/paper/2309.13705/) (2023)

## Built on by (most cited first, 1 of 1)

- [IP-Basis PINNs: Efficient Multi-Query Inverse Parameter Estimation](https://mlsanity.com/paper/2509.07245/) (2025)

## Abstract (arXiv)

In engineering and applied mathematics, developing accurate mathematical models to predict and understand real-world phenomena is of utmost importance. Symbolic regression is a useful machine learning-based tool to fit models but it can be computationally expensive. We present a new method for enhancing symbolic regression for differential equations via dimensional analysis, specifically the Buckingham Pi theorem and Ipsen's method. Since symbolic regression often suffers from high computational costs and overfitting, nondimensionalizing datasets reduces the number of input variables, simplifies the search space, and ensures that derived equations are physically meaningful. As a first step, we combine dimensional analysis with the PySR symbolic regression algorithm to show that dimensional analysis improves the accuracy of recovering algebraic equations. The results demonstrate that transforming data into a dimensionless form significantly improves the training and test error of the symbolic expressions found. Then, as our main contribution, we perform nondimensionalization guided by Ipsen's method. We then incorporate the nondimensionalized equation into a pipeline combining Universal Physics-Informed Neural Networks and symbolic regression to recover the unknown term when a differential equation is only partially known. We find that symbolic regression is able to better recover the unknown term after nondimensionalizing the data, under both noisy and noiseless conditions. These findings suggest that integrating dimensional analysis with symbolic regression can significantly lower computational costs and increase accuracy, providing a robust framework for automated discovery of governing equations in complex systems when data is limited.