Abstract: Traditional experimental-based research methods for hydrogen production via water electrolysis are constrained by factors such as cost, safety, and observational limitations, making it difficult to reveal the microscopic characteristics and interaction mechanisms of multi-physics fields in the electrolysis process. Numerical simulation technology transforms real physical systems into virtual mathematical models, enabling the simulation and behavioral prediction of hydrogen production via water electrolysis. This approach drives advancements in optimizing electrolyzer operating conditions, structural design, and other related areas. The numerical modelling and simulation methods are used to study the proton exchange membrane (PEM) electrolysis cell in this work. A three-dimensional multi-physics model of PEM electrolyzer, considering electrochemical reaction, fluid flow, heat transfer and two-phase flow, is established to simulate the working voltage, electrolysis efficiency, oxygen distribution, and temperature distribution of the electrolyzer under steady-state operation at different input current density conditions. The effects of operating condition including current density, water input temperature and water flow rate on the operation and overpotential of the electrolyzer is quantified. Optimization calculations are performed on the inlet water temperature and flow rate conditions to reduce the operating voltage, and the optimized operating conditions are experimentally validated. The results show that the increase of current density leads to the increase of operating voltage. At low current density of 1.0 A/cm², the polarization losses dominate. At high current density of 2.0 A/cm², the contribution of ohmic and concentration losses is elevated. Increasing the water temperature and the water flow rate both help to reduce the operating voltage. Increasing the water temperature leads to a rise in polarization overpotential and a drop in ohmic overpotential, while the concentration overpotential remains essentially unchanged, resulting in an overall reduction in total overpotential. Conversely, increasing the flow rate causes the concentration overpotential to decrease, with both polarization and ohmic overpotentials remaining relatively stable, ultimately lowering the total overpotential. However, the marginal benefits diminish as both water temperature and flow rate continue to rise. While adjusting the water temperature and flow rate, the minimum operating voltage corresponds to an optimum water temperature of 60 ℃ when the flow rate does not exceed 0.2 m/s; however, when the water flow rate is greater than 0.2 m/s, the optimal water temperature is 70 ℃.