The f(Q) theories of modified gravity arise from the consideration of non-metricity as the basic geometric quantity, and have been proven to be very efficient in describing the late-time Universe. We use the Big Bang Nucleosynthesis (BBN) formalism and observations in order to extract constraints on various classes of f(Q) models. In particular, we calculate the deviations that f(Q) terms bring on the freeze-out temperature $$T_f$$T f in comparison to that of the standard $$\Lambda $$Λ CDM evolution, and then we impose the observational bound on $$ \left| \frac{\delta {T}_f}{{T}_f}\right| $$δ T f T f to extract constraints on the involved parameters of the considered models. Concerning the polynomial model, we show that the exponent parameter should be negative, while for the power-exponential model and the new hyperbolic tangent-power model we find that they pass the BBN constraints trivially. Finally, we examine two DGP-like f(Q) models, and we extract the bounds on their model parameters. Since many gravitational modifications, although able to describe the late-time evolution of the Universe, produce too-much modification at early times and thus fall to pass the BBN confrontation, the fact that f(Q) gravity can safely pass the BBN constraints is an important advantage of this modified gravity class.

We use observational data from the Pantheon supernovae sample, direct Hubble constant measurements with cosmic chronometers, the cosmic microwave background shift parameter ${\mathrm{CMB}}_{\text{shift}}$, and redshift-space distortion ($f{\sigma }_8$) measurements, in order to constrain $f(T)$ gravity. We do not follow the common $\gamma$ parametrization within the semianalytical approximation of the growth rate, in order to avoid model-dependent uncertainties. To our knowledge this is the first time that $f(T)$ gravity has been analyzed within a Bayesian framework, and with background and perturbation behaviour considered jointly. We show that all three examined $f(T)$ models are able to adequately describe the $f{\sigma }_8$ data. Furthermore, by applying the Akaike, Bayesian and deviance information criteria, we conclude that all considered models are statistically equivalent; however the most efficient candidate is the exponential model, which additionally presents a small deviation from the $\mathrm{\Lambda }\mathrm{CDM}$ paradigm.

Physics-Informed Neural Networks (PINNs) have emerged as a robust framework for solving Partial Differential Equations (PDEs) by approximating their solutions via neural networks and imposing physics-based constraints on the loss function. Traditionally, Multilayer Perceptrons (MLPs) have been the neural network of choice, with significant progress made in optimizing their training. Recently, Kolmogorov-Arnold Networks (KANs) were introduced as a viable alternative, with the potential of offering better interpretability and efficiency while requiring fewer parameters. In this paper, we present a fast JAX-based implementation of grid-dependent Physics-Informed Kolmogorov-Arnold Networks (PIKANs) for solving PDEs, achieving up to 84 times faster training times than the original KAN implementation. We propose an adaptive training scheme for PIKANs, introducing an adaptive state transition technique to avoid loss function peaks between grid extensions, and a methodology for designing PIKANs with alternative basis functions. Through comparative experiments, we demonstrate that the adaptive features significantly enhance solution accuracy, decreasing the L2 error relative to the reference solution by up to 43.02%. For the studied forward PDE problems, our methodology approaches or surpasses the results obtained from architectures that utilize up to 8.5 times more parameters, highlighting the potential of adaptive, grid-dependent PIKANs as a superior alternative in scientific and engineering applications.