Machine Can Automatically Discover Parametric Functions to Model HEP Data

arXiv:2607.19750 · hep-ex, cs.LG · Submitted 2026-07-22 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: We're looking at the paper: "Machine Can Automatically Discover Parametric Functions to Model HEP Data".

Jane: The gist The iterative process can be automated by a machine using symbolic regression,

Tom: First, who's behind it and why it matters.

Paper summary: Tom: So, looking at the paper "Machine Can Automatically Discover Parametric Functions to Model HEP Data," the authors are essentially showing how symbolic regression can automate the creation of models for binned data in high-energy physics.

Jane: They demonstrate that this automated process can be used on CMS and ATLAS Run two dijet spectra, finding functions that match existing dijet and UA2 models <ref:2607.19750#pg1,CMS and ATLAS Run 2 dijet spectra>.

Lu: The implication is that we can potentially bypass the tedious manual guesswork of functional form selection in analysis, which is a major bottleneck right now.

Meng: For deployment, I still have questions about the practical implementation details regarding scaling this up to handle real-world data streams efficiently, as you'd need robust infrastructure for those symbolic operations.

Tom: The paper focuses on showing that the method works by demonstrating it on five hundred sixty independent seeded runs across seven configurations, achieving a chi squared over degrees of freedom of approximately one in many cases <ref:2607.19750#pg3>.

Jane: So, while they show success and relevance to existing models like the dijet function, we should keep in mind what they didn't cover.

Lu: They mention that the search is constrained by defining a set of elementary math operators and rules that connect them, which limits the types of functions it can discover to those defined by those rules.

Tom: That’s a key limitation; you can only find functions within the space they've defined, not some completely arbitrary mathematical expression.

Jane: So, for someone listening who isn't in HEP, the big change is seeing how an AI system can be used to automatically discover complex mathematical relationships from raw data without a human having to pre-define those relationships.

Meng: If this approach proves scalable and reliable outside of these specific benchmark spectra, then it could significantly speed up many other kinds of automated modeling tasks in production environments.

Tom: That's the core idea behind this work on "Machine Can Automatically Discover Parametric Functions to Model HEP Data." It’s about automating the discovery part of scientific modeling.

Conclusion: Tom: So, this paper is about using symbolic regression to automatically find mathematical functions that fit messy binned data in high-energy physics, specifically for things like dijet spectra.

Jane: It sounds like they're tackling the decades-old headache of manually guessing a function shape when you only have empirical data and no true underlying formula.

Lu: Exactly. The authors are proposing an automated search over function space, where the math operators and rules themselves become trainable parameters during the fit process.

Meng: But I always wonder about the cost to run this thing in production; does this symbolic search happen fast enough for a real-time analysis pipeline?

Tom: The paper shows they paired their symbolic regression engine, PySR, with an optimizer called LMFIT to handle those exact uncertainty measures needed for physics.

Jane: And the result is pretty neat—they tested it on CMS and ATLAS Run two dijet data and found functions that actually rediscovered known dijet and UA2 models.

Lu: That’s a significant result because it means the AI isn't just fitting random noise; it’s finding mathematically meaningful structures that already exist in the data.

Tom: It shifts the manual work from guessing *what* to do with the function to defining *how* the function can be built, which is a big mechanical change.

Jane: So, what does this mean for someone who just wants to understand this? It suggests that instead of spending weeks tweaking a curve by hand, we could let an AI search millions of mathematical possibilities for us.

Meng: If it works on these benchmark spectra, the question is whether it breaks when the data gets really complex or when you move to a completely different type of physics signal.

Tom: That’s the next thing we need to check—the generalization beyond those specific dijet setups and how robust those discovered functions are.

University of Pennsylvania University of European Organization for Nuclear Research University of Cambridge Institute for Artificial Intelligence and Fundamental Interactions University of Wisconsin-Madison University of California San Diego Massachusetts Institute of Technology Institute for Artificial Intelligence and Fundamental Interactions

hep-ex, cs.LG

Submitted: 2026-07-22

Updated: 2026-10-19

Comments: Proceedings of the 43rd International Conference on High Energy Physics (ICHEP 2026)

Code: https://github.com/hftsoi/symbolfit

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 82/100

The gist: The gist The iterative process can be automated by a machine using symbolic regression, which performs a data-driven search over function space without requiring prior knowledge of what an adequate

Key concepts

Symbolic Regression (SR): A machine learning method that treats the functional form of a model as a 'trainable parameter.' It searches through mathematical operators and rules to dynamically construct candidate functions that best fit the data, eliminating the need for humans to pre-define what an adequate function should look like.
SymbolFit: A package designed to automate three modeling steps: searching for functions using SR, re-optimizing numerical constants with uncertainty estimates via LMFIT, and evaluating candidate functions based on metrics like $\chi^2/NDF$ or RMSE. It streamlines the process from initial guess to a statistically rigorous model.
Functional Form Search: The core search mechanism where the system explores a massive space of possible mathematical expressions connecting elementary operators. This data-driven approach allows the machine to discover complex, non-intuitive functional shapes that might be missed by human intuition alone.
Uncertainty Modeling: Because symbolic regression returns exact functions without inherent error measures, SymbolFit interfaces with LMFIT to re-optimize the numerical constants and calculate a covariance matrix. This step is crucial for HEP analyses as it provides the essential uncertainty information needed for statistical hypothesis testing.

Terminology

Symbolic Regression — 2× in this paper · explained in 6 episodes

Summary

The gist The iterative process can be automated by a machine using symbolic regression, which performs a data-driven search over function space without requiring prior knowledge of what an adequate function should look like.

Introduction

Modeling binned data with smooth functions is ubiquitous in high-energy physics (HEP) data analysis, including deriving smooth scale factors to correct data-simulation mismodeling and modeling signal and background processes for hypothesis testing in new physics searches. The main challenge is that most practical binned distributions do not have a “true” underlying function and hence an adequate functional form cannot be derived from first principles but only empirically. The standard practice has remained the same for decades, even as of today. An analyzer first guesses a functional form by intuition and writes down the parameterization by hand, then fits it to data and examines the residuals. The first trials often fail with bad fit quality as the intuitive functional form is not adequate to describe a nontrivial distribution shape, and one needs to modify the form by, say, adding more parameters here and swapping the math operators there. This repetitive process of form fine-tuning and refitting is empirical and creates a laborious loop that would sometimes impede the analysis progress. The final candidate functions surviving this loop are often confined to a small corner of function space and highly fine-tuned to the analysis at hand. They may no longer work for a future analysis after a luminosity increase or a trigger change, and a new family of functions has to be invented all over again. Well-known examples are the “dijet function” used to model the QCD background in CMS and ATLAS Run 2 high-mass dijet resonance searches [1, 2], and the older “UA2 function” used alongside the dijet function in an ATLAS Run 2 low-mass dijet search [3]. SR is a machine learning method that performs a data-driven search over function space, where the functional form can be viewed as a “trainable parameter” that is dynamically changing during a fit to minimize the residual loss. The function space is defined by a set of elementary math operators and the rules that connect them to construct a candidate function. This eliminates the need to know what an adequate functional form looks like before fitting, as the search itself is automated. We have previously introduced SymbolFit1 [4], an API that interfaces PySR [5] (SR search engine) and LMFIT [6] (nonlinear optimization and uncertainty estimation) for automating parametric modeling designed to target these HEP analysis applications We show that not only does it succeed with a nontrivial rate, but it also readily discovers a wide range of different functions that model the spectra as well as the dijet and UA2 functions.

SymbolFit

SymbolFit streamlines three steps into a single pipeline for automation:.

1 Function search. PySR searches for adequate functions over a function space based on a multi-population evolutionary algorithm, without requiring a predefined functional form.

2 Re-optimization and uncertainty estimation. Functions returned by PySR (and SR algorithms in general) are exact, carrying no uncertainty measure, which is an essential input to statistical analyses in HEP. Hence, the returned functions are parameterized and LMFIT is interfaced to re-optimize the numerical constants in each function and to compute uncertainties via the covariance matrix.

3 Evaluation. All candidate functions are scored (e.g., κ2/NDF, p-value, RMSE, R2)

A search typically returns tens of functions, stochastically drawn from the large function space, and can be repeated with a different random seed to explore different batches of functions.

Rediscovery setup

We use two public datasets available on HEPData: the Run 2 observed dijet mass spectra at √s = 13 TeV published by CMS [7] and ATLAS [8], specifically Figure 5 from Ref. [1] and Figure 3a from Ref. [2], respectively, where the dijet function in Eq. 1 was used to model the spectra in the published results <ref:2607.19750 pg4, we use two public datasets available on HEPData: the Run 2 observed dijet mass spectra at √s = 13 TeV published by CMS [7] and ATLAS [8], specifically Figure 5 from Ref. [1] and Figure 3a from Ref. [2], respectively, where the dijet function in Eq.

Improvements for AI systems

The symbolic regression search can replace manual functional form selection by performing a data-driven search over function space without requiring prior knowledge of what an adequate function should look like. This allows AI systems to automatically discover complex parametric models for binned HEP data, eliminating the laborious loop that would sometimes impede the analysis progress caused by trial-and-error fitting.
The integrated SymbolFit pipeline enables automated uncertainty modeling, as it pairs symbolic regression with uncertainty modeling to target HEP analysis use cases. This means improved AI systems can not only discover functions but also provide essential statistical inputs like the covariance matrix and total uncertainty coverage via parameter sampling for every candidate function.
The system can perform automated hypothesis testing by evaluating candidate functions using metrics such as chi2 /NDF, p-value, RMSE, R2. This allows AI to rapidly screen thousands of potential models, ensuring that the selected function is not only statistically adequate but also rigorously validated against the data before being used in a final analysis.

Abstract

In HEP data analyses, finding an adequate function to model binned data has largely relied on a manual process: guess a functional form by intuition, fit, examine, then repeat until successful. We show that this iterative process can be automated by a machine using symbolic regression, which performs a data-driven search over function space without requiring prior knowledge of what an adequate function should look like. We present the SymbolFit package, which pairs symbolic regression with uncertainty modeling to target HEP analysis use cases, and demonstrate it on the CMS and ATLAS Run 2 dijet spectra: 560 independent seeded runs across seven simple fit configurations generated over 1000 functions fitting the spectra with χ 2/NDF about 1, and 111 of the runs rediscovered the very dijet and UA2 functions used in published dijet searches.