## 4 Discussion  
In analyzing the hot extrusion phenomenon of AA6082 it is rather important that the roles of given (input) and predicted (output) parameters can be easily exchanged by posing the question: what should the chemical content and/or process parameters properties be in order to achieve some prescribed yield strength or elongation? In this case, the same ANN model is applicable, and only the selection of input and output parameters must be changed. More specifically, this means that we can relatively easily find inverse functional laws between the parameters that describe the phenomenon. Mathematically, the problem can be written simply by first defining a general function \(y\), which is a function of \(N\) input parameters \(x_{i}\):  
\[y=f(x_{1},\ x_{2},\ \ldots,x_{i},\ \ldots,\ x_{N}) \tag{6}\]  
The predicted value of any input parameter can now be defined as:  
\[\hat{x}_{j}=f(y,\ x_{1},\ x_{2},\ \ldots,x_{i},\ \ldots,\ x_{N});i\
eq j \tag{7}\]  
In a simplified case, when we want to predict, using basic notation, an input parameter \(\hat{x}_{j}\) that is in relation to two input parameters (\(x_{2},\ x_{3}\)) at fixed values of the output parameter \(y\), the above expression can be written as:  
\[\hat{x}_{1}=f(\ x_{2},\ x_{3},\ y=const) \tag{8}\]  
An illustration of the above expression (Equation (8)) is shown in Figure 7, where the prediction for Si content (\(\hat{x}_{1}\)) as a function of ram speed (\(x_{2}\)) and Mn content (\(x_{3}\)) at a fixed yield strength (\(y\)) which amounts to 322 MPa is presented. An automated explanation of the results in Figure 7 reads as:  
_These two graphs present the relationship between the manganese content (Mn [wt.%]) on the \(y\)-axis, the ram speed (nm/s) on the \(x\)-axis, and the predicted silicon content (Si) in weight percent, as well as the precision of this prediction (indicated by standard deviation), while maintaining a fixed parameter of yield strength at 322 MPa._  
**Figure 7a (Solid Lines): Predicted Silicon Content**The solid lines represent isolines of predicted silicon content (Si) in the material. Each solid line connects points with the same predicted silicon content, with the numbers on the lines indicating the specific Si value. This graph indicates how varying Mn content and ram speed can influence the predicted silicon content in the material when the yield strength is held constant at 322 MPa. Figure 7b (**Dotted Lines: Standard Deviation of Predicted Silicon Content**  
The dotted lines in this graph show the standard deviation of the predicted silicon content. The numbers along these lines represent the magnitude of the standard deviation; lower numbers indicate a more precise prediction. A higher standard deviation value indicates less confidence in the predicted silicon content at a given combination of Mn content and ram speed. Using the Graphs with Fixed Yield Strength Parameter: Since yield strength is fixed at 322 MPa, the graphs show how the silicon content varies with Mn content and ram speed while keeping yield strength constant. This constraint provides insight into the specific effects of Mn content and ram speed on silicon content without the complicating factor of varying yield strength. To analyze the graphs: Select a Mn content and ram speed combination to analyze. Check the graph to determine the predicted Si value at that point. Refer to the graph (Figure 7b) to assess the standard deviation for the prediction to understand the confidence level. Example Analysis: Let's take an example point at a Mn content of 0.48 wt.% and a ram speed of 15 mm/s. In the graph Figure 7a, this might fall near a contour line labeled \"0.885\", suggesting a predicted silicon content of 0.885 wt.%. In the graph (Figure 7b), the same point might lie near a line labeled \"0.081\", which would indicate a standard deviation in the prediction of the silicon content of 0.081 wt.%.