## 1 Introduction  
Nimonic 80A, as a nickel-based superalloy, has been widely used in jet engines for aircraft, gas turbines for power plant and marine diesel engines because of its high creep strength, superior oxidation resistance and strong resistance to corrosions at high temperature [1; 2; 3]. Generally, the Nimonic 80A is used to fabricate exhausting valve. The upsetting and closed die forging are traditionally applied to form the exhausting valve. However, in recent years, the electric upsetting process with isostatic loading and high heating efficiency is developed to form the exhausting valves [4; 5]. It is well known that the hot deformation behavior for a specific material is sensitive to the hot deformation parameters involving strain, strain rate and temperature, and is highly non-linear during hot deformation. The flow behavior of materials is often complex due to the comprehensive function of hardening and softening mechanisms. Consequently, modeling and prediction of the constitutive stress-strain relationships with a high precision is quite complex in nature; meanwhile, it is significant to study and understand the hot deformation behavior and furthermore optimize the deformation process (electric upsetting, forging and extrusion) by numerical simulations. How to obtain an accurate strain-stress relationship becomes critical for the correct calculation of the finite element model [6]. So far, a large amount of research on the characterization for complex non-linear relationships between true stress and deformed parameters such as strain, strain rate and temperature at elevated temperatures has been proposed. Numerous efforts have been made to three types of constitutive models involving the analytical constitutive model, phenomenological constitutive model and artificial neural network [7]. In analytical models, constitutive relations are derived based on physical theories, which require very clear understanding of the processes that control the deformation of the materials. The phenomenological constitutive model is an accurate mathematical model and has relatively many coefficients that need to be calibrated with experimental data. A phenomenological model including the Arrhenius-type equation with hyperbolic laws was proposed to predict flow stress [8]. Furthermore, an improved Arrhenius-type constitutive model incorporating the strain effect on the hot deforming parameters, has been developed to describe and predict the flow behavior for diverse materials or alloys. [PERSON] _et al._ proposed a modified hyperbolic sine constitutive equation, in which the influence of strain was incorporated to predict the flow stress of 42 CrMo steel [9]. Later, the modified Arrhenius-type equation was precise for describing the elevated temperature flow stress of Aermet100 steel [10], Ti60 titanium alloy [11], Al-Zn-Mg-Er-Zr alloy [12], _etc._ Such constitutive equations are typically only applicable to the limited materials with specific conditions due to the poor adaptability for the new experimental data. Additionally, the artificial neural network (ANN) model with a back-propagation learning algorithm has been successfully used to predict the hot working behavior of material to overcome the gross approximations introduced by the regression methods [6; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. The back-propagation artificial neural network (BP-ANN) is a model emulating some functions of biological neural networks with a data-driven black-box structure [24], thus it merely needs a collection of some typical examples from the anticipant mapping functions for training regardless of explicit professional knowledge of deformation mechanisms. The BP-ANN model with a data-driven black-box provides a novel way to predict the flow stress by learning the complex and non-linear relationships of flow stress, strain rate, strain and temperature with true stress-strain data. [PERSON] _et al._ applied a feed-forward back-propagation ANN model to predict the flow stress of Aermet100 steel [13].