## 1 Introduction  
Particulate matter in the air with aerodynamic diameters <2.5 \(\mu\)m (PM\({}_{2.5}\)) not only threaten the quality of local atmospheric environments, but also the long-term regional and global climate [1; 2; 3; 4; 5; 6]. The dynamics of the ultrafine particles, such as corrosion and weathering of materials, might impact human health, especially in megacities [7; 8; 9; 10; 11]. From previous studies, PM\({}_{2.5}\) concentrations can be estimated by two traditional approaches, deterministic and statistical methods [12; 13; 14; 15; 16; 17; 18; 19; 20]. The former is based on the theory of atmospheric dynamics, applying physical and chemical processes of atmospheric pollutants to establish a numerical model for prediction. Its accuracy depends on how accurate the sources of pollution, the real-time emissions and the description of the physical and chemical processes [21; 22]. In contrast, the statistical method establishes the prediction models using a large number of existing historical data, based on statistical methods, such as linear regression analysis and nonlinear machine learning. Generally, statistical models are more efficient for discovering the relationship between pollutants and potential predictors, yet lack explicit description of physical mechanisms [23; 24; 25; 26; 27; 28; 29]. In recent years, machine learning methods have brought new tools to the prediction of PM\({}_{2.5}\) concentration, e.g., neural networks. [PERSON] et al. (2013) optimized a hidden Markovmodel (HMM), which significantly reduced the false alarm rate of PM\({}_{2.5}\) concentration in Northern California [30]. [PERSON] et al. (2015) used a neural network to predict PM\({}_{2.5}\) concentration through the observations of O\({}_{3}\), CO, PM\({}_{10}\) and other air pollutants [25]. They found that the accuracy of the neural network model is better than multivariate regression models. A deep autoencoder (DA) and a long short-term memory (LSTM) model was used to predict PM\({}_{10}\) and PM\({}_{2.5}\), and both DA and LSTM models performed well [14]. Due to the complexity of the PM\({}_{2.5}\) variability, hybrid models with multiple neural networks have been proposed recently, which usually perform better than a single model. [PERSON] et al. (2018) introduced a GM-ARMA hybrid model to forecast short-term PM\({}_{2.5}\) conditions [31]. Compared to the single GM or ARMA model, the GM-ARMA model achieved a better fitting. [PERSON] et al. (2018) constructed a hybrid model of one-dimensional convolution and LSTM [9]. They used the PM\({}_{2.5}\) concentration, accumulated wind speeds and precipitation as predictors, and obtained a better prediction skill than the single LSTM model. [PERSON] et al. (2018) combined a random forest neural network and a back-propagation (BP) neural network [21]. [PERSON] et al. (2020) used a CPSO-BP neural network [24]. They adopted the Chaos Particle Swarm Optimization (CPSO) algorithm to update the initial weights and thresholds of the BP neural network. Most of these studies showed that the prediction of a hybrid model is better than a single model. One of the challenges for PM\({}_{2.5}\) prediction is that the forecasting errors tend to increase during heavily polluted days, which is a common problem for machine learning methods. As is known, during heavily polluted days the PM\({}_{2.5}\) concentrations can be influenced by multi-scale processes from hourly to annual variability, such as weather conditions, pollution sources and emissions, which presents a tough challenge for machine learning. We know that a single machine learning model can perform very well for single-frequency time sequences but is not good at learning multi-frequency sequences. In this paper, an innovative idea to solve this problem is proposed: a multi-frequency sequence of the target time series can be decomposed into a set of intrinsic mode functions (IMF). For each IMF, we can use different machine learning methods to obtain the maximal learning efficiency and, therefore, the best forecast skill.