## 1 Introduction
Compared with existing heuristic algorithms (such as genetic algorithm (GA), particle swarm optimization (PSO), and ant colony algorithm), FOA has few parameters to adjust, and it is easy to understand and implement by virtue of the simple computational process. Moreover, it has many advantages such as less control variables, strong control ability, and good global optimization. The optimization solves these problems and has been used by many scholars to solve complex parameter optimization problems [22; 23; 24]. However, when a fruit fly in the fruit fly population finds the optimal solution for this iteration, all the fruit flies gather at the location of the fruit fly, but if the fruit found by the fruit fly is not the global optimal solution, then the algorithm falls into local optimum. Based on this, this paper proposes a variational mode decomposition based on the improved fruit fly algorithm (IFOA), which can not only adaptively optimize the internal parameters K and \(\alpha\) of VMD but also improve the convergence speed and convergence precision of the algorithm. How to extract fault characteristic information from nonlinear time series is the key to fault diagnosis of high-pressure common rail injector. In recent years, with the development of nonlinear scientific theories, various information entropy methods have emerged. [PERSON] [25] proposed the concept of approximate entropy in 1991. Then, for the self-matching defect of approximate entropy, [PERSON] et al. [26] proposed the concept of sample entropy. Sample entropy is a commonly used feature extraction method, which has the advantages of strong anti-noise ability and short time series, but the method fault feature states can only be described from a single scale. [PERSON] et al. [27; 28] proposed a multi-scale entropy (MSE) based on sample entropy to measure the complexity of time series at different scales. Aiming at the sample entropy similarity measure in MSE, the mutation occurs easily. [PERSON] et al. [29], combined with the concept of fuzzy entropy, proposed multiscale fuzzy entropy (MFE) and applied it to the fault diagnosis of rolling bearings. In order to extract the fault information of high-frequency components in the signal, [PERSON] [30] introduced the concept of hierarchical fuzzy entropy. Compared with multi-scale fuzzy entropy, hierarchical fuzzy entropy considers both the low-frequency component and the high-frequency component of the signal, thus providing more comprehensive and accurate time mode information. In order to alleviate the shortcomings of sample entropy, fuzzy entropy, and permutation entropy, [PERSON] (2018) proposed dispersion entropy (DE) and proposed to measure the complexity of time series from different scale factors. Multiscale Dispersion Entropy (MDE) ([PERSON] and [PERSON], 2016) does not need to rank the amplitude values of one embedded vector, nor does it need to calculate the distance between any two composite delay vectors in different embedding dimensions. Compared with sample entropy and fuzzy entropy, dispersion entropy has the advantages of simple and fast calculation ([PERSON] and [PERSON], 2016; [PERSON], 2016). At the same time, dispersion entropy overcomes the main defects of permutation entropy and effectively solves the influence of medium amplitude of embedding vector ([PERSON] and [PERSON], 2016). The study of analog signals and biosignals shows that dispersion entropy is relatively insensitive to noise and excellent in anti-interference compared to sample entropy and fuzzy entropy because small changes in amplitude values do not change their class labels; dispersion entropy is more sensitive to changes in the synchronization frequency, amplitude value, and signal bandwidth. Therefore, based on the superiority of analytic hierarchy process and dispersion entropy, hierarchical dispersion entropy (HDE) based on hierarchical entropy and dispersion entropy is proposed.