⚛️ high-energy experiments

Machine Can Automatically Discover Parametric Functions to Model HEP Data

This paper introduces SymbolFit, a machine learning package that automates the discovery of parametric functions for modeling High Energy Physics data through symbolic regression, successfully replicating known dijet spectra functions and achieving excellent statistical fits without relying on manual intuition.

Original authors: Ho Fung Tsoi, Dylan Rankin, Cecile Caillol, Miles Cranmer, Sridhara Dasu, Javier Duarte, Philip Harris, Elliot Lipeles

Published 2026-07-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ho Fung Tsoi, Dylan Rankin, Cecile Caillol, Miles Cranmer, Sridhara Dasu, Javier Duarte, Philip Harris, Elliot Lipeles

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 a detective trying to solve a mystery, but the clues aren't fingerprints or footprints; they are tiny, jagged bumps on a graph. This is the daily life of a physicist working in High-Energy Physics (HEP), the field where scientists smash particles together at nearly the speed of light to see what the universe is made of. When these particles collide, they create a chaotic spray of debris. To make sense of this chaos, physicists have to organize the debris into "bins" (like sorting marbles by size) and then draw a smooth, curvy line through the dots to represent the data. This line is called a "function," and it acts like a map, helping scientists distinguish between ordinary background noise and something truly new and exciting, like a hidden particle.

For decades, drawing this map has been a very human, very frustrating job. It's like trying to guess the shape of a secret object by feeling it through a thick blanket. A physicist has to take a wild guess at what the curve should look like, draw it, check if it fits, and if it doesn't, tweak the guess, draw it again, and repeat. This "guess-and-check" loop can take forever, and the final shape often feels more like a lucky guess than a solid discovery. But what if a machine could do the guessing for you? What if an algorithm could explore millions of mathematical shapes in seconds to find the perfect fit without ever needing to know what the answer looks like beforehand? That is the big question this paper asks, and the answer might change how physicists do their work forever.

The Machine That Learns to Draw Curves

In this study, a team of researchers built a digital detective named SymbolFit to see if a computer could automatically discover the best mathematical formulas to model particle collision data. Instead of asking a human to guess the shape of the curve, they let the machine explore a vast "forest" of possible mathematical functions. Think of it like giving a robot a bag of Lego bricks (mathematical operators like plus, minus, multiply, divide, and exponents) and asking it to build a structure that perfectly matches a pile of scattered data points. The robot doesn't know what the final building should look like; it just tries millions of combinations until it finds one that fits the data perfectly.

The researchers tested this machine on real data from two giant particle detectors, CMS and ATLAS, which have been smashing protons together at the Large Hadron Collider. Specifically, they looked at the "dijet spectra," which are graphs showing how often pairs of particle jets appear at different energy levels. These graphs are notoriously tricky to model, and for years, physicists have relied on two specific, hand-crafted formulas known as the "dijet function" and the "UA2 function" to describe them. These formulas were discovered by humans through years of trial and error.

The team set up an experiment with 560 independent runs of their machine, using seven different search configurations (some letting the robot use any math it wanted, others giving it a specific template to follow). They wanted to see two things: Could the machine rediscover the famous human-made formulas on its own? And could it find other formulas that worked just as well?

The results were surprisingly successful. In 111 of the 560 runs, the machine independently "rediscovered" the exact same formulas that human physicists had spent years deriving. It found the dijet function and the UA2 function purely by looking at the data, without being told what to look for. But here is the most exciting part: the machine didn't just copy the humans. It found over 1,000 different candidate functions that fit the data just as well as the famous human formulas. In fact, the rediscovered human formulas were just a tiny fraction of the total successful shapes the machine found.

The paper shows that these new, machine-discovered functions are not just mathematical curiosities; they are robust. The researchers used a method called uncertainty modeling to ensure that every curve the machine drew came with a built-in "safety margin" (a measure of how confident the machine is in its fit). They found that many of these new shapes were completely different from the old ones—some used different combinations of logs and exponents, while others used entirely new structures—but they all described the particle data with a high degree of accuracy (a statistical score called $\chi^2$ /NDF ≈ 1).

The authors suggest that this approach could replace the old, laborious method of humans guessing and tweaking formulas. Instead of a physicist spending weeks fine-tuning a single curve, a machine could instantly generate a whole library of perfect-fitting functions, giving scientists more options and saving them from the "guess-and-check" loop. While the paper doesn't claim this is a solved problem for every single physics question, it strongly suggests that for modeling particle data, the future might belong to machines that can automatically discover the math behind the mystery.

Technical Summary: Machine Can Automatically Discover Parametric Functions to Model HEP Data

Problem Statement
In High-Energy Physics (HEP) data analysis, modeling binned data with smooth functions is a ubiquitous task, ranging from deriving scale factors for data-simulation corrections to modeling signal and background processes for new physics searches. The primary challenge is that most practical binned distributions lack a "true" underlying function derivable from first principles; adequate functional forms must be determined empirically. The current standard practice relies on a laborious, manual iterative loop: an analyst guesses a functional form based on intuition, fits it to data, examines residuals, and manually modifies the form (e.g., adding parameters or swapping operators) until a satisfactory fit is achieved. This process is empirical, creates a repetitive loop that can impede analysis progress, and often results in candidate functions that are highly fine-tuned to specific analysis conditions (e.g., specific luminosity or trigger settings), potentially failing in future analyses. Well-known examples of such empirically derived functions include the "dijet function" and the "UA2 function" used in CMS and ATLAS Run 2 dijet resonance searches.

Methodology
The authors propose using Symbolic Regression (SR) to automate the discovery of parametric functions. Unlike traditional regression where the functional form is fixed and only parameters are optimized, SR treats the functional form itself as a "trainable parameter" that dynamically changes during the fit to minimize residual loss. The search occurs over a function space defined by a set of elementary mathematical operators and the rules connecting them.

To implement this for HEP, the authors utilize SymbolFit, a package that integrates:

PySR: A multi-population evolutionary algorithm search engine that explores the function space without requiring a predefined functional form.
LMFIT: A nonlinear optimization and uncertainty estimation tool. Since SR algorithms return exact functions without uncertainty measures (essential for HEP statistical analyses), SymbolFit re-optimizes the numerical constants of the returned functions and computes uncertainties via the covariance matrix.

The study evaluates SymbolFit using a "rediscovery" experiment. The goal was to determine if a data-driven search could rediscover the empirically derived dijet and UA2 functions (Eq. 1) using only the published Run 2 dijet mass spectra from CMS and ATLAS ( $\sqrt{s} = 13$ TeV).

The experimental setup involved seven distinct search configurations (summarized in Table 1 of the paper), ranging from "free-form" spaces with broad operator sets (e.g., $\{+, -, \times, /, \wedge, \text{exp}, \text{ln}, \text{tanh}\}$ ) to "template" configurations that impose specific structural constraints (e.g., $f = \exp(h(x))$ ). Each configuration was run 40 times with different random seeds on both the CMS and ATLAS datasets, totaling 560 independent seeded runs. A run was considered a "rediscovery" if it produced at least one candidate algebraically equivalent to the dijet or UA2 function families with a fit quality of $\chi^2/\text{NDF} < 2$ .

Key Results

Rediscovery Success: Out of 560 runs, 111 runs successfully rediscovered either the dijet function or the UA2 function. No single run rediscovered both simultaneously.
- Even nearly assumption-free searches (e.g., the ops-minimal configuration) found the functions, though at lower rates (e.g., 3/40 and 2/40 runs for CMS and ATLAS, respectively).
- When search templates were imposed to narrow the building blocks toward the known function structures (e.g., tpl-exp-log), rediscovery rates increased significantly (e.g., 36/40 and 24/40 runs for CMS and ATLAS).
Alternative Function Discovery: Rediscovery was not the sole outcome. The search generated over 1,000 candidate functions across all runs that achieved $\chi^2/NDF \approx 1$ . The rediscovered dijet and UA2 functions constituted only a tiny fraction of these successful candidates.
Diversity of Forms: The study identified a wide variety of distinct functional forms (listed in Table 2 and plotted in Figure 3) that modeled the spectra as well as the traditional dijet and UA2 functions. These included forms with different operator combinations and structures, demonstrating that the "true" functional form is not unique.
Uncertainty Modeling: Unlike raw SR outputs, every candidate function returned by SymbolFit included embedded uncertainty estimation. The authors demonstrated this by plotting candidates with 68% and 95% uncertainty bands propagated from the parameter covariance matrix (Figure 4).

Significance and Claims
The paper claims that the traditional, manual, trial-and-error approach to selecting adequate functional forms in HEP can be replaced and automated by a machine using data-driven symbolic regression. The results demonstrate that:

Machines can autonomously rediscover complex, empirically derived functions used in major HEP analyses without prior knowledge of their form.
The search space contains a vast diversity of functions that fit the data equally well, suggesting that the specific functional forms currently used in HEP are not unique solutions but rather one of many adequate possibilities.
The SymbolFit package successfully bridges the gap between symbolic regression and HEP statistical requirements by providing necessary uncertainty modeling.

The authors conclude that this automated, data-driven approach can be broadly applied to any HEP data analysis scenario where parametric modeling is performed, potentially reducing the laborious nature of model selection and increasing the robustness of analysis pipelines against changes in experimental conditions.

The Machine That Learns to Draw Curves

Technical Summary: Machine Can Automatically Discover Parametric Functions to Model HEP Data

More like this