## 1 Introduction  
As a product of the transition from traditional fuel to electric vehicles (EVs), plug-in hybrid electric vehicles (PHEVs) are currently a hot topic in research on new energy vehicles. As one of the keys to energy-saving technologies for hybrid electric vehicles, energy management strategies (EMS) can be divided into three main approaches: 1 Rule-based (RB) energy management strategies [1; 2]; 2 optimization-based control strategies; and 3 intelligent control-based algorithm strategies. RB strategies commonly used in the research field include the charge-depleting-charge sustaining (CD-CS) strategy and fuzzy logic (FL)-based strategy. A rule-based strategy is simple and can be easily implemented in real time using look-up tables, although the fuel economy is relatively poor. Optimization-based EMS can be divided into instantaneous and global optimization. A representative of an instantaneous optimization algorithm is the equivalent fuel consumption minimization strategy (ECMS) [3; 4; 5], which achieves a good real-time performance, but cannot be operated stably under different working conditions. Global optimization algorithms mainly include dynamic programming (DP) [6; 7], Pontryagin's minimum principle (PMP) [8; 9], etc. Although global optimization algorithms can guarantee the minimum fuel consumption, all driving cycle information is needed and the calculation cost is also large. Therefore, these algorithms cannot be used for online control. In contrast, based on machine learning (ML), artificial neural networks (ANNs), and other intelligent algorithms, the global optimization algorithm can be applied online by fitting the optimal control law obtained by the algorithm [10]. At present, a large number of studies have realized the online application of a global optimization algorithm in a hybrid electric vehicle energy management strategy through a neural network. In [11; 12], the authors used the battery power and engine speed as control variables, and applied dynamic programming to calculate the global optimal fuel consumption. To obtain the best battery power of the hybrid electric vehicles (HEV), as well as the best engine speed, the neural network is trained using the vehicle speed, required power, battery state of charge (SOC), and driving style of the offline calculation results. In addition, another neural network module based on \(11\) variables has been used to predict the operating mode. This method accurately fits the calculation results of dynamic programming; however, the calculation cost of dynamic programming is significant, the neural network module is too bloated, and the complexity affects the stability of online applications. In [13], it is shown that online rules extracted from trained neural networks are difficult to adapt to different driving conditions. A large number of current studies have improved the generalization of energy management strategies by improving the neural network algorithm; however, they have rarely started from the improvement of optimization algorithms to improve the robustness of the neural network. To a certain extent, the complexity of the optimization algorithm determines the robustness of the online control strategy. This study therefore improves the accuracy of neural network recognition by improving the dynamic programming algorithm. The calculation cost of dynamic programming mainly depends on the dimension of the state variables and control parameters, as well as the degree of dispersion in each dimension. For power-split hybrid vehicles or vehicles with gearboxes, the torque and speed are decoupled, and because they are all continuous variables, the transmission system has two degrees of freedom. To reduce the complexity of dynamic programming, in [14], the best point of operation efficiency according to different engine powers of hybrid vehicles was found when applying a power-split configuration, and the speed corresponding to this point was used as the engine operating point under the current power.