## 1 Introduction  
In ultra-high performance concrete (UHPC) mixture design, the compressive strength, and the flexural strength are two key mechanic parameters to evaluate the strength and the ductility of materials. Because UHPC is brittle, the steel fibers with high tensile strength and high ultimate elongation, are always uniformly dispersed in UHPC to increase the ductility and the strength of concrete. The UHPC with steel fibers is also called ultra-high performance steel fiber reinforced concrete (UHPFRC). The steel fibers in UHPFRC improve the cohesive forces between fibers and matrix, change the granular skeleton, and increase the anchorage length between fibers and the surrounding matrix [1; 2]. Besides, steel fibers bridge cracks and retard the propagation to increase the strength and the ductility of UHPFRC [3]. Unfortunately, too many steel fibers lead to fibers inter-wrap and interlock with each other, affecting the workability of UHPFRC, to reduce the strength of UHPFRC [4]. Furthermore, steel fibers are expensive and numerous steel fibers added to the UHPFRC cost too much. Thus, predicting the compressive strength and the flexural strength of UHPFRC accurately can optimize mix proportion, control the volume fraction of steel fibers, and decrease the costs of UHPFRC. However, evaluating the flexural strength and the compressive strength of the UHPFRC is a huge challenge due to the complex composite behavior caused by the properties steel fibers (diameter (\(D\)), length (\(L\)), aspect ratio (\(AR\)), and volume fraction (\(VF\))) and concrete matrix (water to binder ratio (\(W/B\)) and concrete strength without fibers). Nowadays, the contribution of cement-based materials to sustainability is a topic of study [5; 6; 7; 8] and the performance of several additions in cement-based materials, such as silica fume, fly ash, the water to cement ratio, and so on, has been analyzed, which could be also suitable for being used in high performance concrete. This fact also makes it necessary to consider their influence on the mechanical properties of these new concretes. In the past decades, different experimental and theoretical investigations [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30] have been carried out on the compressive strength and the flexural strength of UHPFRC. Several analytical models have been constructed to predict the compressive strength and the flexural strength of UHPFRC by proposing a series of assumptions about the mechanism and developing the equilibrium equations. Furthermore, several empirical models [31; 32; 33; 34; 35; 36], including those used in the design codes, have been established based on the experimental studies to predict the compressive strength and the flexural strength of UHPFRC. Additionally, the Chinese standard JGJ/T 221 [31] only proposes the empirical model to predict the flexural strength of steel fiber reinforced concrete. In general, these models mainly depend on the diameter, the length, and the volume fraction of steel fibers and the compressive/flexural strength of UHPFRC without fibers. However, the experimental data used to establish the empirical formula is limited, and most of them mainly focus on the volume fraction of steel fibers and does not consider other parameters. Thus, when new test data is available, the predictive accuracy and reliability of those empirical models have to be re-evaluated. In recent years, the artificial intelligence techniques of artificial neural networks (ANNs) have been widely adopted to solve many engineering complex problems due to its high accuracy and adaptability. The ANN model has been successfully used to evaluate the early-age autogenous shrinkage of concrete [37], estimate the compressive strength of FRP-confined (FRP denotes fiber reinforced polymer) concrete circular columns [38], optimize the mix compositions of steel fiber-reinforced concrete [39], and so on.