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Abstract:

In cosmology, emulators play a crucial role by providing fast and accurate predictions of complex physical models, enabling efficient exploration of high-dimensional parameter spaces that would be computationally prohibitive with direct numerical simulations. Symbolic emulators have emerged as promising alternatives to numerical approaches, delivering comparable accuracy with significantly faster evaluation times. While previous symbolic emulators were limited to relatively narrow prior ranges, we expand these to cover the parameter space relevant for current cosmological analyses. We introduce approximations to hypergeometric functions used for the Λ cold dark matter (ΛCDM) comoving distance and linear growth factor which are accurate to better than 0.001% and 0.05%, respectively, for all redshifts and for Ωm∈[0.1,0.5]. We show that integrating symbolic emulators into a Dark Energy Survey Year 1 (DES-Y1)-like 3×2 pt analysis produces cosmological constraints consistent with those obtained using standard numerical methods. Our symbolic emulators offer substantial improvements in speed and memory usage, demonstrating their practical potential for scalable, likelihood-based inference. This article is part of the discussion meeting issue 'Symbolic regression in the physical sciences'.

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 focused on the promising regions in order to be equally performant. This article is part of the discussion meeting issue 'Symbolic regression in the physical sciences'.

Abstract:

If dark matter annihilates into standard model particles with a cross section which is velocity dependent, then Local Group dwarf galaxies will not be the best place to search for the resulting gamma ray emission. A greater flux would be produced by more distant and massive halos, with larger velocity dispersions. We construct full-sky predictions for the gamma ray emission from galaxy- and cluster-mass halos within $∼200\mathrm{Mpc}$ using a suite of constrained $N$ -body simulations () based on the Bayesian Origin Reconstruction from Galaxies algorithm. Comparing to observations from the Large Area Telescope and marginalizing over reconstruction uncertainties and other astrophysical contributions to the flux, we obtain constraints on the cross section which are 2 (7) orders of magnitude tighter than those obtained from dwarf spheroidals for $p$ -wave ( $d$ -wave) annihilation. We find no evidence for either type of annihilation from dark matter particles with masses in the range $m_{\mathrm{χ}}=2\mathrm{–}500\mathrm{GeV}/c^2$ , for any channel. As an example, for annihilations producing bottom quarks with $m_{\mathrm{χ}}=10\mathrm{GeV}/c^2$ , we find $a_1<2.4×{10}^{−21}{\mathrm{cm}}^3{\mathrm{s}}^{−1}$ and $a_2<3.0×{10}^{−18}{\mathrm{cm}}^3{\mathrm{s}}^{−1}$ at 95% confidence, where the product of the cross section, $\mathrm{σ}$ , and relative particle velocity, $v$ , is given by $\mathrm{σ}v=a_{ℓ}(v/c{)}^{2ℓ}$ and $ℓ=1$ , 2 for $p$ - and $d$ -wave annihilation, respectively. Our bounds, although failing to exclude the thermal relic cross section for velocity-dependent annihilation channels, are among the tightest to date.

Abstract:

Abstract Symbolic regression (SR) has emerged as a powerful method for uncovering interpretable mathematical relationships from data, offering a novel route to both scientific discovery and efficient empirical modelling. This article introduces the Special issue on symbolic regression for the physical sciences, motivated by the Royal Society discussion meeting held in April 2025. The contributions collected here span applications from automated equation discovery and emergent-phenomena modelling to the construction of compact emulators for computationally expensive simulations. The introductory review outlines the conceptual foundations of SR, contrasts it with conventional regression approaches and surveys its main use cases in the physical sciences, including the derivation of effective theories, empirical functional forms and surrogate models. We summarize methodological considerations such as search-space design, operator selection, complexity control, feature selection and integration with modern AI approaches. We also highlight ongoing challenges, including scalability, robustness to noise, overfitting and computational complexity. Finally, we emphasize emerging directions, particularly the incorporation of symmetry constraints, asymptotic behaviour and other theoretical information. Taken together, the papers in this Special issue illustrate the accelerating progress of SR and its growing relevance across the physical sciences. This article is part of the discussion meeting issue ‘Symbolic regression in the physical sciences’.

Abstract:

Abstract The peculiar velocity field of the local Universe provides direct insights into its matter distribution and the underlying theory of gravity, and is essential in cosmological analyses for modelling deviations from the Hubble flow. Numerous methods have been developed to reconstruct the density and velocity fields at z ≲ 0.05, typically constrained by redshift-space galaxy positions or by direct distance tracers such as the Tully–Fisher relation, the fundamental plane, or Type Ia supernovae. We introduce a validation framework to evaluate the accuracy of these reconstructions against catalogues of direct distance tracers. Our framework assesses the goodness-of-fit of each reconstruction using Bayesian evidence, residual redshift discrepancies, velocity scaling, and the need for external bulk flows. Applying this framework to a suite of reconstructions—including those derived from the Bayesian Origin Reconstruction from Galaxies (BORG) algorithm and from linear theory—we find that the non-linear BORG reconstruction consistently outperforms others. We highlight the utility of such a comparative approach for supernova or gravitational wave cosmological studies, where selecting an optimal peculiar velocity model is essential. Additionally, we present calibrated bulk flow curves predicted by the reconstructions and perform a density–velocity cross-correlation using a linear theory reconstruction to constrain the growth factor, yielding S8 = 0.793 ± 0.035. The result is in good agreement with both weak lensing and Planck, but is in strong disagreement with some peculiar velocity studies.