# A Topological Approach to Enhancing Consistency in Machine Learning via Recurrent Neural Networks
## 4 Experiments
### Real-Life Example  
While the adaptability of the proposed consistency-based RNN transition formula in Equation (4) allows for its application and testing across diverse real-world datasets in different engineering disciplines, our study specifically concentrates on enhancing localization estimation by learning from the deformation history of sheet metal. This focus was chosen due to its significance in understanding and predicting material behavior under varying conditions, a critical aspect in the field of material science and engineering  
**Forming Limit Curve (FLC) Estimation**: Forming Limit Curves (FLCs) are essential in materials science, particularly in the field of sheet metal forming. They draw the boundary between safe deformation and failure in metals, such as localization or tearing. Developed through empirical methods, FLCs graphically plot the major and minor strains a material can endure before failing. This graphical representation is crucial for optimizing manufacturing processes, informing material selection, and ensuring quality in industries like automotive and aerospace. In the study [42], the authors defined a functional mapping in strain space for learning localization points from the deformation history of sheet metal through machine learning methods. Their theoretical hypothesis was tested using a bilinear loading dataset [42] generated by numerical simulations with an FE-based Marciniak-Kuczynski (MK) model. They trained various ML algorithms, and NN architectures on this dataset, successfully estimating three different bilinear FLCs. However, they used a fixed number of 400 increments for the loading paths in the training and estimation process. On the other hand, due to the computational nature of FE-based numerical simulations, any given loading path can be represented by thousands or millions of increments, depending on the temporal frequency of the information extracted from the simulation. Therefore, a trained NN architecture should be able to consistently estimate localization points with different numbers of increments for the same loading path, not just the fixed 400 increments used in this study. To test our proposed consistency solution, shown in Equation (4), for the estimation of bilinear FLCs, we simulated three different FLC cases shown in Figure 7 using the same MK model in the study by [42]. The time-frequency of the numerical simulation is set to obtain 100,000 increments for each bilinear loading path. Two different NN architectures are built, consisting of a GRU layer with 128 neurons and a ConsRNN layer with 256 neurons. While the GRU layer uses sigmoid, the ConsRNN layer uses tanh as the recurrent activation function. In both architectures, the recurrent layers are followed by a fully connected layer with a sigmoid activation function. Unlike the study by [42], we did not use the loading direction \(\phi\) as an input, so we only used the principal and minor strains as inputs to the NN architectures to learn the mapping from the loading history and the amount of deformation. The NN architectures are trained for 100 epochs on the bilinear loading dataset, and while the GRU-built architecture gave \(7.3\times 10^{-4}\) of MSE, the ConsRNN-built architecture gave \(3.3\times 10^{-3}\) of MSE on the test set. The trained architectures are used to obtain localization estimates for 100,000 incremented bilinear paths by substituting them in 400, 1000 and 10,000 increments with intervals of 250, 100 and 10. From the results shown in Figure 9, for three different FLC cases with different numbers of increments, it can be observed that the predicted curves from the NN architecture built by ConsRNN are consistent as opposed to the NN architecture built by GRU. Although the estimates are more accurate for 400 increments using the architecture built by GRU, the estimated curves move away from the ground truth as the number of increments increases.