## 1 Introduction  
Granular mixture is widely applied in industrial manufacturing, agricultural production and civil engineering, including powder metallurgy, food accumulation and building materials. For these mixtures, porosity is one of the most crucial factors that has a considerable impact on mixture structure and performance. However, due to the complex particle composition, it is hard to calculate or predict the mixture porosity without the help of lab experiments. In order to solve the above problems, many researchers have proposed different methods to calculate and analyze the porosity of granular mixtures. In particuology, packing density calculation models based on the packing theory were derived, including the Furnas model [1], linear packing density model [2], compressible packing model [3] and other improved models [4; 5]. All of them are analytical models which were used to analyze particle spatial distributions and mixture packing densities. However, some literature [6; 7] confirmed that these models were not accurate enough for complex problems and where particles have a wide range of sizes. A typical mixture in pavement engineering, however, is mainly composed of mineral aggregates with a wide range of particles sizes. The fine aggregates are less than 2.36 mm while the large particles can be larger than 19.0 mm. Additionally, in pavement engineering, mixtures are formed through the lab or field compacting process which is dependent on the particle physical interactions. Evidently, these analytical models cannot consider the particle interactions during the compacting process. Based on the two considerations above, the authors believe that the conventional analytical models as discussed above are not applicable for predicting porosities of pavement mixtures. In pavement engineering, experimental testing methods were popularly used for studying mixture porosities [8; 9]. In the past decades, many research studies were conducted for building the relationship between porosities and compositions of mixtures, while most of them were applied to specific projects or limited conditions [10; 11]. Since 2005, with the rapid development of the discrete element methods (DEM), virtual mixture models have been developed for studying mix porosities [12; 13], where particle gradations and morphological features were considered. With the DEM-based study, the researchers tended to explore the relations between the mix porosity and its mix composition. However, in the existing literature, the research purposes were mainly focused on evaluating a limited number of mixtures instead of finding a universal model for predicting mixture porosities [14; 15]. In the past decades, machine learning algorithms have been widely used to solve problems which have multiple variables with no clear relations. Various models have been developed in industrial design [16], disease diagnosis [17], engineering analysis [18], material optimization [19; 20; 21], etc., which validated the high accuracy and the universal application of machine learning algorithms. Back Propagation Neural Network algorithm (BPNN), which is a multilayer feedforward neural network trained according to the error back propagation algorithm, is one of the most widely used algorithms. There are some advantages to this method: strong nonlinear mapping ability, highly self-learning and adaptive ability, good generalization ability and certain fault-tolerant ability. In recent years, some research efforts have been made in asphalt pavement engineering to study asphalt mixture's volumetric properties, gradations and asphalt contents with the BPNN [22; 20; 23]. Additionally, the BPNN-based methods were also used for the prediction of cement concrete compression strength [24; 25; 26]. As demonstrated in the literature, the BPNN was useful and efficient in solving asphalt or concrete pavement problems. Obviously, the BP models have advantages for predicting mixture porosities.