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Symbolic Regression in Scientific Machine Learning: From Data Noise to Governing Equations

Introduction

In recent years, machine learning has become widely used in scientific contexts, producing strong predictive results. However, an increasingly clear limitation has emerged: many models perform well numerically but remain difficult to interpret.

At ActarusLab, our work focuses on this gap. We explore whether it is possible, starting from complex datasets, to recover simple and interpretable mathematical structures that describe the underlying system.

The problem with black-box models

Many modern machine learning approaches, particularly deep neural networks, are highly effective but difficult to analyze.

In practice, this means:

• they provide accurate predictions
• but offer limited insight into the underlying mechanisms

This becomes a critical issue in domains such as quantitative finance, physics, and computational chemistry, where interpretation is an essential part of validation.

Symbolic regression as an alternative approach

Unlike traditional machine learning methods, symbolic regression does not assume a fixed functional form.

Instead, it searches directly for a mathematical relationship of the form:

The goal is not only to minimize prediction error, but to identify expressions that are:

• interpretable
• verifiable
• stable across different conditions

High-fidelity simulation

In many cases, the process begins with the construction of high-fidelity simulations that approximate the behavior of the system under study.

These environments allow controlled data generation, which is essential for testing hypotheses in a reproducible way.

Extracting mathematical structure

The methodology can be summarized in three stages:

A simple example

In some cases, the resulting model can take a form such as:

What makes this interesting is not its complexity, but the fact that it is:

• directly interpretable
• testable on new data
• usable without an underlying black-box model

Applications

Interpretability as a scientific constraint

A model that cannot be explained is difficult to validate rigorously.

For this reason, in our approach interpretability is not an optional feature but a constraint.

We prioritize models that are simpler and verifiable, even at the cost of some predictive performance.

Conclusion

The objective is not to replace scientific reasoning with more complex algorithms, but to use algorithms to recover simpler and more understandable representations of complex systems.

In this sense, symbolic regression provides a bridge between data complexity and mathematical structure.