# A Topological Approach to Enhancing Consistency in Machine Learning via Recurrent Neural Networks
## 1 Introduction  
In machine learning (ML), \"consistency\" encompasses several key aspects. It involves the ability of a model to yield stable and reliable results across diverse datasets and conditions, reflecting good generalization and absence of overfitting [1, 2, 3, 4, 5, 6]. Theoretically, it relates to an algorithm's increasing accuracy with larger training datasets, aligning with asymptotic unbiasedness. Algorithmic consistency is about reproducibility of results under uniform conditions [7, 8, 9, 10, 11, 12]. In data handling, it signifies data uniformity and reliability, crucial for accurate model outputs [13, 14, 15]. Lastly, in model evaluation, consistency implies the use of standardized metrics for fair and meaningful performance assessment [16, 17, 18, 19]. This paper focuses on a less-explored aspect of consistency. Namely, we focus on scenarios where an event in a sequential dataset is influenced by past occurrences, meaning that current events are affected by events that have already occurred, or in other words, current events are history-dependent. The events in a sequence represent snapshots of an ongoing evolutionary process taken at a certain time interval. We claim that current MLmethods can learn these sequences and make valid predictions only when the predictive model inputs have the same discretization as that used in the training phase. In contrast, when the discretization length changes, the model gives incorrect, or inconsistent, results. Therefore, the consistency problem arises from the challenge of representing history-dependent problems using a discrete set of data points when we are unable to control the discretization length. As machine learning becomes increasingly integrated into various engineering applications involving physical processes, we believe this issue will grow in importance. One area where consistency becomes crucial is in the numerical finite element (FE) modeling of the mechanical behavior of components and structures. In these FE simulations, the non-linear response of the structure or structural component is resolved incrementally by discretizing the entire event into millions of time increments. Machine learning models have been used recently in these simulations as metamodels carrying the information about material behavior at the microscale. Thereby, metamodels bring the local material response to structural scale simulations with considerably reduced computational effort compared to traditional computational homogenization methods. However, the success of the entire endeavor is dependent on the consistency between discretization length (time step size) and training phase and its application phase in structural scale simulations. If the time increments in solving the mechanical problem are different between those two phases, the ML based metamodel will not produce accurate results. Hence, we reach the same argument as in [20], who stated, _\"The predictions of a mechanical model should depend on the loading path itself and not on a particular discretization thereof\"_. In this study, we explore consistency within RNNs, inspired by recent works of [20,21]. While these studies focus on consistency in RNNs for constitutive modeling, we believe the problem requires further investigation, particularly considering the purely mathematical nature of the problem. The aim of this work is to introduce a specialized RNN formula that provides consistent estimates across various real-world applications. Our proposed approach introduces a simple but effective RNN formula that delivers consistent estimates across different real-world applications. We demonstrate the effectiveness of our approach through experiments on both synthetic and real-world datasets. The main contributions of this paper can be summarized as follows:  
* We introduce a specialized RNN transition function known as the Consistent RNN Cell (ConsRNN), which is designed to ensure consistent estimates as the number of samples within the same domain increases during testing.