By varying the intrinsic initial geometry, the p/d/$^3$He+Au collisions at the Relativistic Heavy Ion Collider (RHIC) provide a unique opportunity to understand the collective behavior and probe the possible sub-nucleon fluctuations in small systems. In this paper, we employ the hybrid model \vishnu~ with \trento~ initial conditions to study the collective flow and the fluid behavior in p/d/$^3$He+Au collisions. With fine-tuned parameters, \vishnu~ can describe the $v_2(p_T)$ and $v_3(p_T)$ data from the PHENIX and STAR collaborations. However, for some certain parameter sets with initial sub-nucleon fluctuation, the hydrodynamic simulations have already beyond their limits with the average Knudsen number $\langle K_n \rangle$ obviously larger than one. Our calculations demonstrate that, for a meaningful evaluation of the fluid behavior in the small systems, model simulations should also pay attention to the validity range of hydrodynamics.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.

In this proceeding, we will briefly review our recent progress on implementing deep learning to relativistic hydrodynamics. We will demonstrate that a successfully designed and trained deep neural network, called stacked U-net, can capture the main features of the non-linear evolution of hydrodynamics, which also rapidly predicts the final profiles for various testing initial conditions.

In this paper, we investigate the Kibble-Zurek scaling of the sigma field and net-protons within the framework of Langevin dynamics of model A. After determining the characteristic scales $\tau_{kz},l_{kz}$ and $\theta_{kz}$ and properly rescaling the traditional cumulants, we construct universal functions for the sigma field and approximate universal functions for net-protons in the critical regime, which are insensitive to the relaxation time and the chosen evolving trajectory. Besides, the oscillating behavior for the higher order cumulants of net-protons near the critical point is also drastically suppressed, which converge into approximate universal curves with these constructed Kibble-Zurek functions.

In this proceeding, we will briefly review our recent progress on implementing deep learning to relativistic hydrodynamics. We will demonstrate that a successfully designed and trained deep neural network, called stacked U-net, can capture the main features of the non-linear evolution of hydrodynamics, which could also rapidly predict the final profiles for various testing initial conditions.