Symbolic Regression: Bridging Interpretability and Complexity in Machine Learning

Introduction

Some of the most pivotal scientific discoveries—like Kepler's third law of planetary motion and Planck's law—emerged from symbolic relationships derived from data. In these cases, the simplicity of the underlying equations brought clarity and allowed further discoveries. However, finding such relationships in today’s high-dimensional datasets can seem like an impossible task without the use of automated tools.

Enter Symbolic Regression (SR)—a method that automates the discovery of interpretable mathematical expressions from data. SR uses evolutionary algorithms to explore a vast space of possible equations, searching for those that best fit the data. With the rise of neural networks and their often opaque "black-box" nature, symbolic regression offers a path to AI explainability—making complex models more understandable.

We’ll explore what symbolic regression is, how it works, and how it plays a crucial role in scientific discovery and AI explainability. Additionally, we'll dive into some tools like PySR, which further enhance symbolic regression, and look at Symbolic Distillation, a method that distills neural networks into interpretable mathematical expressions.

What is Symbolic Regression?

Symbolic regression is a machine learning technique that searches for the best-fitting mathematical equation to describe a dataset, without assuming any prior form. Unlike traditional regression models (e.g., linear or polynomial regression), symbolic regression can discover novel relationships from data by building expressions from a set of basic mathematical operators.

For example, rather than fitting data to a predetermined equation like y=mx+b, symbolic regression can derive an equation such as y=sin(x)+log(z) depending on what best describes the data.

• How Equations Are Represented:In symbolic regression, equations are constructed from basic operators such as addition (+), subtraction (-), multiplication (*), division (/), powers, trigonometric functions (e.g., sin, cos), and logarithmic functions (log).These expressions are represented as trees, with each node representing an operator (e.g., + or *) and each branch representing a variable or constant. This allows the symbolic regression algorithm to explore a wide variety of equation forms.

How Symbolic Regression Works: Genetic Programming

Symbolic regression often uses genetic programming—an evolutionary algorithm inspired by natural selection—to find the best-fitting equations. Just like biological evolution, symbolic regression evolves equations over time by using operations like mutation and crossover.

• Mutation: Mutation involves randomly modifying part of an equation to explore new possibilities. For example, in a formula like y=x+2, mutation might change the constant 2 to 3, or replace the addition with multiplication.
• Crossover: Crossover is the process of combining parts of two different equations to form a new one. If one equation is y=x+sin(z) and another is y=log(w), crossover might combine them into y=x+log(w).

These operations help explore the vast space of possible equations, with fitness functions evaluating each equation's ability to explain the data accurately. Over generations, the best-performing equations are selected and refined.

Symbolic Distillation: Making Neural Networks Explainable

Miles Cranmer, the creator of the PySR library, introduces the concept of Symbolic Distillation, which aims to make neural networks interpretable by distilling their complex behavior into simple, symbolic expressions.

• What Is Symbolic Distillation? Symbolic Distillation is the process of using symbolic regression to approximate the function learned by a neural network. Essentially, this technique creates a symbolic representation (a human-readable equation) that mimics the neural network's behavior.This approach allows users to reverse-engineer neural networks and gain insights into the relationships they have learned, making the model's decision-making process clearer and more interpretable.

For example, rather than leaving a neural network to explain its decisions as a black box, symbolic distillation can provide an equation like y = x1^2  + sin(x2) offering a clear, mathematical description of the neural network's logic.        

The Broader Context: Interpretable AI

The black box problem in machine learning refers to the opacity of many models—particularly deep neural networks—that perform well but offer little insight into their inner workings. Interpretability is essential in fields like healthcare, law, and finance, where the reasoning behind AI decisions must be transparent and explainable.

While symbolic regression is one promising approach to interpretability, it is not the only one:

• LIME (Local Interpretable Model-agnostic Explanations): LIME explains the decisions of any black-box model by perturbing the input and observing the changes in output.
• SHAP (Shapley Additive Explanations): SHAP values provide a way to quantify the contribution of each feature in a model's output, offering insight into why a model made a particular decision.

While both LIME and SHAP are popular, they only provide local explanations (for specific instances), while symbolic regression provides global explanations by generating a model that is interpretable across all data points.

Advantages of Symbolic Regression

• Interpretability: The output of symbolic regression is a simple, human-readable equation, making it highly interpretable compared to complex neural networks.
• Scientific Discovery: Symbolic regression has been used to rediscover known physical laws and identify new patterns in data, offering potential for breakthroughs in fields like physics and biology.
• Flexibility: Symbolic regression does not assume a specific equation form. It builds equations dynamically based on the data, making it more flexible in a wide range of domains.

Challenges and Limitations

• Computational Complexity: Searching through the vast space of possible equations can be computationally expensive, particularly as datasets grow in size.
• Scalability: Symbolic regression can struggle with large, high-dimensional datasets compared to more traditional machine learning methods.
• Risk of Overfitting: The algorithm may find overly complex equations that fit the training data perfectly but do not generalize well to unseen data.

Other Tools Supporting Symbolic Regression

While PySR is a powerful tool for symbolic regression, several other libraries also offer similar capabilities:

• gplearn: A genetic programming library for symbolic regression in Python, designed to work with scikit-learn models.
• Eureqa: A commercial software platform specifically designed for symbolic regression, often used in scientific discovery and engineering applications.
• TuringBot: An AI-driven symbolic regression tool that allows for automated model discovery without the need for human intervention.

Real-World Applications

Symbolic regression has already shown promise in several real-world applications:

• Predictive Maintenance in Manufacturing: By identifying key relationships between sensor data and machine failures, symbolic regression can provide interpretable models that predict when equipment is likely to fail.
• Scientific Discovery: Symbolic regression has rediscovered physical laws and helped uncover hidden relationships in datasets, from chemistry to astrophysics.
• Financial Market Analysis: In finance, symbolic regression can help identify the underlying factors driving market trends, providing transparent and interpretable models for trading algorithms.

Future Directions

• Improving Efficiency and Scalability: Ongoing research aims to make symbolic regression more scalable, particularly in handling large datasets common in modern machine learning.
• Integration with Deep Learning: Hybrid models combining symbolic regression with neural networks offer the potential to combine the power of deep learning with the interpretability of symbolic equations.
• Automated Scientific Discovery: Symbolic regression could be the key to automating scientific discovery, uncovering new laws and relationships from raw data.

Conclusion

As the demand for AI interpretability grows, symbolic regression offers a compelling solution to bridge the gap between powerful machine learning models and human understanding. Whether through symbolic distillation for neural networks or rediscovering scientific laws, symbolic regression provides interpretable, transparent models that can explain complex phenomena.

By combining symbolic regression with other AI techniques, we can build hybrid models that retain both predictive power and explainability, paving the way for more trustworthy and transparent AI systems.