## 1 Introduction
The common probability density estimation method is no longer easy to use; many scholars have put forward improvement measures, including [PERSON]'s efficient sample density estimation by combining gridding and an optimized kernel [15], [PERSON]'s fast nonparametric conditional density estimation [16], [PERSON]'s accurate photometric redshift probability density estimation [17], [PERSON]'s novel composite quantile regression neural network (CQRNN) model [18], [PERSON]'s hybrid wind power probability density prediction method based on quantile regression neural network and Epanechnikov kernel function using unbiased cross-validation [19], and so on. These methods can reduce disturbance or similar items in the observation samples when the kernel density estimation model is subjected to correlation processing, filtering, clustering or compressing subsets. Finally, the above methods can reduce the training time of probability density estimation and improve the training efficiency of the model. In these previous studies, [PERSON] proposed quantile regression (QR) to solve the problems dealing with the asymmetric distribution of response variables or the scattered distribution of data using ordinary least square (OLS) [20]. Quantile regression (QR) can explain the effects of input variables on response variables at different quorum, and it can also obtain more information about the relationship between input variables and response variables. What is more, quantile regression can not only reveal the heterogeneous relationship between input variables and response variables at each quorum but also obtain the conditional quantile information of input variables at continuous quorum, which is convenient for scientific decision making. However, the QR proposed by [PERSON] is still based on linear regression. The result of this processing can describe and reflect the simple linear relationship between those variables, but it is difficult to deal with the complex nonlinear functions. [PERSON] proposed a method that is a hybrid of KPCA, alpha-regression quantile and adaptive genetic algorithm (AGA). This method overcomes the limitation of Kernel principal component regression (KPCR) and achieves better prediction accuracy [21]. Building on the aforementioned research, most probabilistic forecasts of transportation demand assume that the timing errors of aircraft entering and exiting airspace or airports follow a normal distribution. Alternatively, historical statistical values of traffic flow are often defaulted to a certain normal distribution pattern from the perspective of uncertainty. However, an analysis of currently available actual operational data revealsthat in many cases, this normal distribution either does not exist or is limited to only a few ideal operating days. Therefore, to address potential shortcomings in traditional deterministic prediction methods, this paper proposes a probabilistic traffic demand prediction approach that integrates various methods from the perspective of demand uncertainty. The approach combines neural networks with the quantile regression method. By leveraging the extremely strong nonlinear adaptive ability of neural networks and the precision of quantile regression in describing explanatory variables, we can obtain 100 consecutive quantiles for the future. Subsequently, using these continuous conditional quantiles, the continuous probability density function and the probability density curve of a certain day in the future can be realized through the kernel density estimation method. This approach not only provides specific point prediction values and their change intervals but also calculates the probability of each value within the prediction change interval for en route sector traffic demand. It allows for a more accurate point prediction value for a given day, effectively correcting biases in the understanding of probabilistic traffic demand.