M. P. Bento

M. P. Bento, H. B. Câmara, J. R. Rocha et al.

Stiff differential equations pose a major challenge for Physics-Informed Neural Networks (PINNs), often causing poor convergence. We propose a simple, hyperparameter-free method to address stiffness by normalizing loss residuals with the Jacobian. We provide theoretical indications that Jacobian-based normalization can improve gradient descent and validate it on benchmark stiff ordinary differential equations. We then apply it to a realistic system: the stiff Boltzmann equations (BEs) governing weakly interacting massive particle (WIMP) dark matter (DM). Our approach achieves higher accuracy than attention mechanisms previously proposed for handling stiffness, recovering the full solution where prior methods fail. This is further demonstrated in an inverse problem with a single experimental data point - the observed DM relic density - where our inverse PINNs correctly infer the cross section that solves the BEs in both Standard and alternative cosmologies.

M. P. Bento, H. B. Câmara, J. F. Seabra

We parametrically solve the Boltzmann equations governing freeze-in dark matter (DM) in alternative cosmologies with Physics-Informed Neural Networks (PINNs), a mesh-free method. Through inverse PINNs, using a single DM experimental point -- observed relic density -- we determine the physical attributes of the theory, namely power-law cosmologies, inspired by braneworld scenarios, and particle interaction cross sections. The expansion of the Universe in such alternative cosmologies has been parameterized through a switch-like function reproducing the Hubble law at later times. Without loss of generality, we model more realistically this transition with a smooth function. We predict a distinct pair-wise relationship between power-law exponent and particle interactions: for a given cosmology with negative (positive) exponent, smaller (larger) cross sections are required to reproduce the data. Lastly, via Bayesian methods, we quantify the epistemic uncertainty of theoretical parameters found in inverse problems.