Abstract: Driven by the low accuracy problem of existing distribution network state estimation (SE) methods when measurements are limited, a Bayesian SE method is proposed for distribution networks based on a novel power flow-informed neural network (PFINN). Firstly, the two-dimensional Gaussian mixture probability distribution of real and reactive power injection is learned for each node using historical data; thereby, abundant samples can be obtained for neural network training by Monte Carlo sampling and power flow calculation. Then, with the goal of minimizing the SE error and power flow equation violation, a Bayesian SE model is established for distribution networks based on the PFINN. Physics loss penalty is introduced into the loss function to constrain the output to be consistent with system operating constraints. Furthermore, the BOHB method is adopted to optimize the hyperparameters of the neural network, while transfer learning is introduced to adapt to changes of network topologies and on-load tap changers. Finally, test results using field data and balanced/unbalanced distribution networks show that the proposed method has better estimation accuracy than the pseudo-measurement-based SE method and the Bayesian SE method without power flow informing. Meanwhile, the proposed method achieves good adaptation performance to topology and tap changes.