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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:

One of the most pressing problems in current cosmology is the cause of the Hubble tension. We revisit a two-rung distance ladder, composed only of Cepheid periods and magnitudes, anchor distances in the Milky Way, Large Magellanic Cloud, NGC4258, and host galaxy redshifts. We adopt the SH0ES (Supernovae and H0 for the Equation of State of dark energy) data for the most up-to-date and carefully vetted measurements, where the Cepheid hosts were selected to harbour also Type Ia supernovae. We introduce two important improvements: a rigorous selection modelling and a state-of-the-art density and peculiar velocity model using Manticore-Local, based on the Bayesian Origin Reconstruction from Galaxies (borg) algorithm. We infer H0 = 71.7 ± 1.3 km s^-1 Mpc^-1, assuming the Cepheid host sample was selected by supernova magnitudes. However, the actual selection criteria are not clear, and other assumptions can increase H0 by up to one statistical standard deviation. The posterior has a lower central value and a 45 percent smaller uncertainty than a previous study using the same distance-ladder data. The result is also slightly lower than the supernova-based SH0ES inferred value of H0 = 73.2 ± 0.9 km s^-1 Mpc^-1, and is in 3.3σ tension with the latest cosmic microwave background results in the standard cosmological model. These results demonstrate that a measurement of H0 of sufficient precision to weigh in on the Hubble tension is achievable using second-rung data alone, underscoring the importance of robust and accurate statistical and velocity-field modelling.

Abstract:

Claims of local ($z \lesssim 0.05$) anisotropy in the Hubble constant have been made based on direct distance tracers such as Tully–Fisher galaxies and Type Ia supernovae. We revisit these using the CosmicFlows-4 Tully–Fisher W1 subsample, 2MTF and SFI++ Tully–Fisher catalogues, and the Pantheon+ supernova compilation (all restricted to $z < 0.05$), including a dipole in either the Tully–Fisher zero-point or the standardized supernova absolute magnitude. Our forward-modelling framework jointly calibrates the distance relation, marginalizes over distances, and accounts for peculiar velocities using a linear-theory reconstruction. We compare the anisotropic and isotropic model using the Bayesian evidence. In the CosmicFlows-4 sample, we infer a zero-point dipole of amplitude $0.087 \pm 0.019$ mag, or $4.1\pm 0.9$ percent when expressed as a dipole in the Hubble parameter. This is consistent with previous estimates but at higher significance: model comparison yields odds of $877\!:\!1$ in favour of including the zero-point dipole. In Pantheon+ we infer zero-point dipole amplitude of $0.049 \pm 0.013$ mag, or $2.3\pm 0.6$ percent when expressed as a dipole in the Hubble parameter. However, by allowing for a radially varying velocity dipole, we show that the anisotropic zero-point model captures local flow features (or possibly systematics) in the data rather than an actual linearly growing effective bulk flow caused by anisotropy in the zero-point or expansion rate. Crucially, inferring a more general bulk flow curve we find results fully consistent with expectations from the standard cosmological model.

Abstract:

The evolution of galaxies is known to be connected to their position within the large-scale structure and their local environmental density. We investigate the relative importance of these using the underlying dark matter density field extracted from the Constrained Simulations in BORG (CSiBORG) suite of constrained cosmological simulations. We define cosmic web environment through both dark matter densities averaged on a scale up to 16 Mpc , and through cosmic web location identified by applying DisPerSE to the CSiBORG haloes. We correlate these environmental measures with the properties of observed galaxies in large surveys using optical data (from the NASA-Sloan Atlas) and 21-cm radio data (from ALFALFA). We find statistically significant correlations between environment and colour, neutral hydrogen gas () mass fraction, star formation rate, and Sérsic index. Together, these correlations suggest that bluer, star-forming, rich, and disc-type galaxies tend to reside in lower density areas, further from filaments, while redder, more elliptical galaxies with lower star formation rates tend to be found in higher density areas, closer to filaments. We find analogous trends with the quenching of galaxies, but notably find that the quenching of low-mass galaxies has a greater dependence on environment than the quenching of high-mass galaxies. We find that the relationship between galaxy properties and the environmental density is stronger than that with distance to filament, suggesting that environmental density has a greater impact on the properties of galaxies than their location within the larger-scale cosmic web.

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.