Symbolic Regression and Differentiable Fits in Beyond the Standard Model Physics

TL;DR

The CMSSM's high-dimensional parameter space makes full-physics evaluations of observables computationally expensive. The authors apply symbolic regression to learn analytic expressions for $m_{H^0}$, $\Omega_{\rm DM} h^2$, and $\delta(g-2)_{\mu}$ as functions of the four CMSSM inputs $m_{1/2}$, $m_0$, $A_0$, and $\tan\beta$, plus a viability classifier, enabling rapid global fits. They demonstrate that SR-constructed expressions reproduce posterior distributions comparable to traditional package-based analyses while reducing computation by about two orders of magnitude. Additionally, the paper develops differentiable SR workflows, replacing non-differentiable operators with differentiable surrogates to enable gradient-based inference via No-U-Turn Sampler (NUTS) and shows that differentiable SR can outperform non-differentiable SR in stability and speed. A direct comparison with neural networks indicates SR offers superior sample efficiency and robust global behavior, while NN performance improves with targeted data. These results highlight SR as a practical, interpretable tool for BSM phenomenology and model comparison, with reusable expressions suitable for updating analyses as new data arrive.

Abstract

We demonstrate the efficacy of symbolic regression (SR) to probe models of particle physics Beyond the Standard Model (BSM), by considering the so-called Constrained Minimal Supersymmetric Standard Model (CMSSM). Like many incarnations of BSM physics this model has a number (four) of arbitrary parameters, which determine the experimental signals, and cosmological observables such as the dark matter relic density. We show that analysis of the phenomenology can be greatly accelerated by using symbolic expressions derived for the observables in terms of the input parameters. Here we focus on the Higgs mass, the cold dark matter relic density, and the contribution to the anomalous magnetic moment of the muon. We find that SR can produce remarkably accurate expressions. Using them we make global fits to derive the posterior probability densities of the CMSSM input parameters which are in good agreement with those performed using conventional methods. Moreover, we demonstrate a major advantage of SR which is the ability to make fits using differentiable methods rather than sampling methods. We also compare the method with neural network (NN) regression. SR produces more globally robust results, while NNs require data that is focussed on the promising regions in order to be equally performant.

Figures (4)

• Figure 1: The performance of the SR expressions for the observables, on the test set. In the upper panels we show the "true vs. prediction" scatter plots. The solid lines delineate the physically viable values. In the lower panels we show the relative errors. Here the vertical lines delineate relative errors of 1%, 10%, and 100%.
• Figure 2: The classifier symbolic regressor performance, showing the output of the classifier when acting on the test data on the left and the ROC curve on the right.
• Figure 3: Posterior probability distributions for the CMSSM found using Dynesty, fitting the three observables $m_{H^0}$, $\delta (g-2)_\mu$, and $\Omega_{\rm DM} h^2$. The package-based results are the scatter plots (and the black lines on the diagonal plots). The SR expression-based results are the red lines. Dimensionful parameters are in GeV. As is customary the 1D marginals are normalized with respect to their maximum heights rather than areas in order to aseess the agreement of maximum a posteriori probabilities.
• Figure 4: Posterior distribution of CMSSM parameters obtained using NUTS compared to those from Dynesty. The contours show the 68% and 95% confidence intervals. The red contours compute the outputs exactly, whereas the remaining ones use the expressions of Ref. AbdusSalam:2024obf (black), a neural network (green), or refined symbolic expressions for a narrower prior range (blue) to bypass this calculation. The 1D marginals are normalized with respect to their maximum heights rather than areas in order to assess the agreement of maximum a posteriori probabilities.