# Physics-Guided Neural Network-Based Feedforward Control for Seamless Pipe Manufacturing Process
## 3 Results
### Neural Network Training
#### 3.1.1 Benchmark Neural Network  
The core concept of this benchmark model is to calculate the necessary valve opening based on the current piston speed and the desired actuation force.  
To establish this relationship, the following test setup is proposed.  
* Both grippers (Figure 3) are engaged, creating a rigid mechanical connection between the two pistons.
* The first cylinder operates in velocity mode, moving at a constant speed.  
Figure 10: Hardware setup for real-time implementation of the PGNN, showing the integration of the PLC and the NPU module.  
* The second cylinder operates in force mode, simulating resistance to the movement of the first cylinder.
* Experiments are conducted for the relevant ranges of piston speeds and forces.  
The basic idea is to train the network on the relevant dataset to achieve the high-quality approximation of the velocity-force characteristics of the actuation system.  
Table 4 presents the achieved valve opening for different force values and a few characteristic piston velocities. The complete dataset used to train the benchmark neural network is provided in the Supplementary Materials (BenchmarkNeuralNetwork_Dataset). It should be noted that the data arrangement in the Supplementary Materials is the same as in Table 4, i.e., the first column contains the force values, the first row lists the velocities, and the rest of the table consists of the valve opening values for each force-velocity combination.  
Considering the need for the real-time implementation of the neural network, a relatively simple network structure was selected.  
* Two inputs: actual piston speed [mm/s] and desired force [kN].
* Two hidden layers with 32 nodes each.
* ReLU activation functions.
* One output: valve opening [%].  
To train the network, the Levenberg-Marquardt backpropagation algorithm [34] was used, with the training parameters shown in Table 5.  
Figure 11 illustrates the training process, showing the loss curve over 500 epochs, which was found to be sufficient as the network achieved a validation result of \(2.2147\times 10^{-9}\), indicating an extremely low error rate and a highly accurate model fit.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Force [kN]**} & \multicolumn{5}{c}{**Piston Velocity [mm/s]**} \\ \cline{2-6}  & **10** & **20** & **30** & **40** & **50** \\ \hline
1 & 3.982 & 6.882 & 9.782 & 12.682 & 15.582 \\
2 & 4.062 & 6.962 & 9.862 & 12.762 & 15.662 \\
3 & 4.142 & 7.042 & 9.942 & 12.842 & 15.742 \\
4 & 4.222 & 7.122 & 10.022 & 12.922 & 15.822 \\
5 & 4.302 & 7.202 & 10.102 & 13.002 & 15.902 \\
6 & 4.382 & 7.282 & 10.182 & 13.082 & 15.982 \\
7 & 4.462 & 7.362 & 10.262 & 13.162 & 16.062 \\ \hline \end{tabular}
\end{table}
Table 4: Valve opening for different force and piston velocity values. Only the part of the dataset used to train the benchmark neural network is presented. The complete dataset is provided in the Supplementary Materials (BenchmarkNeuralNetwork_Dataset).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Minimum Gradient & \(1\times 10^{-7}\) \\ \(\mu\) & 0.001 \\ \(\mu\) Decrease Ratio & 0.1 \\ \(\mu\) Increase Ratio & 10 \\ \hline \end{tabular}
\end{table}
Table 5: Training parameters for the benchmark neural network.