# A Topological Approach to Enhancing Consistency in Machine Learning via Recurrent Neural Networks
## 5 Conclusions  
In this paper, we emphasized the critical importance of consistency in estimation, particularly in sequence-to-sequence learning where the evolution of one continuous event affects another. Focusing on this problem, we proposed a specialized RNN transition formula capable of providing consistent predictions as the number of input samples increases  
Figure 10: As can be seen from the figure (**A**), while the number of increments increasing the MSE for the GRU-trained model is increasing, the trained ConsRNN-based NN architecture still gives similar results, even with the changing number of increments. within the same domain. This theoretical hypothesis was validated through experiments on both synthetic and real-world datasets. The introduced RNN transition formula, termed ConsRNN, was implemented within an NN architecture and compared with a standard GRU-based NN architecture.The architectures are trained on synthetic datasets generated in 1D, 2D, and 3D spaces with 20 input points, and predictions were evaluated across varying increments: 100, 1000, 10,000, and 1,000,000. The results, shown in Figures 4 and 5, demonstrate that while GRU predictions diverge from the ground truth with increasing increments, ConsRNN predictions remain consistent and preserve the topological structure of the ground truth. This behavior is evident in Figure 6, where the GRU predictions (green line) deviate significantly with higher increments, while the ConsRNN predictions (blue line) closely align with the ground truth. Similarly, Figure 8 confirms these findings. When applied to geometric transformations, as shown in Figure 8, ConsRNN consistently preserves the ground truth shape, unlike GRU-based predictions, which deviate as increments increase. These experiments highlight that the specialized ConsRNN formula delivers consistent estimates, whereas standard GRU architectures struggle in various domains. In addition to synthetic datasets, we tested our approach on real-world datasets, as shown in Figures 9 and 10. One application involved predicting forming limit curves (FLCs), which are crucial in materials science for determining the strain limits of sheet metals. The problem is time-dependent, and the data points extracted from simulations are history-dependent due to the nature of finite element simulations. We trained both ConsRNN and GRU architectures on simulation data with 400 increments and evaluated the trained architectures by using 1000, 10000, and 100,000 increments for predictions. Figure 9 demonstrates that while GRU predictions fail to capture the ground truth forming limit curve as the increments increase, ConsRNN predictions remain consistent with the curve. These results show the potential of the proposed transition formula for real-world applications. Another example focused on constitutive modeling, where the inputs were strain and the outputs were stress, using the dataset published in the study by [PERSON] and [PERSON] [20]. The GRU and ConsRNN-based NN architectures were trained on strain inputs with 125 increments, and stress predictions were obtained for 250, 500, 1000, 2000, and 5000 increments. The MSE results in Figure 10 show that GRU performance deteriorates significantly as the number of increments increases, while ConsRNN consistently gives stable and accurate predictions. Example stress predictions for 2000 and 5000 increments further confirm these results. We included the results from traditional methods, specifically the Gated Recurrent Unit (GRU), to show its inability to provide consistent predictions. The main reason for presenting GRU results is its mathematical structure, which inherently does not support consistent estimates. While GRU is widely used in surrogate modeling within materials science, this limitation highlights the novelty and importance of our proposed approach. By contrasting our method with GRU, we aim to emphasize how our formula successfully overcomes this critical challenge.