# Long Short-Term Memory Neural Network Applied to Train Dynamic Model and Speed Prediction
## 2 Related Work  
As mentioned above, parametric approaches of the train dynamic model can be divided into two classes. For the first class, [PERSON] [21] pioneered a single-mass model, where the train is controlled using a finite sequence of traction phases and a final brake phase. The Lagrangian multiplier method is applied to search fuel-efficient driving solutions in this work. To minimize energy consumption, in [22], the authors presented an analytical solution to obtain the sequence of optimal controls. In this calculation algorithm, the in-train force is ignored. With the purpose of shortening the computation time, [PERSON] et al. [23] formulated the single-mass dynamic model as a mixed-integer linear programming (MILP) problem. By approximating and relaxing the high-order nonlinear terms, the global optimal solution can be found in polynomial time. Single-point train control models have also been studied [24; 25]. For the second category, to describe the internal relationship among the adjacent cars, [PERSON] et al. [26] first put forward a multi-point train control model, where a switching policy and linear suboptimal controllers are designed to model this large-scale system. In [27], a multi-point based longitudinal train model was proposed, where the model parameters are tuned according to the experimental environment. This model was validated against actual data from Spoornet. In [28], the authors established a cascade-mass-point model to simplify the multi-point model. Taking simplicity, cost-effectiveness and implementation convenience into consideration,the issue of output regulation with measurement feedbacks is well solved in this model. Besides, many researchers pay attention to building an appropriate multi-point dynamic model in high-speed railways [10; 29; 30]. It is worth mentioning that some data-driven algorithms have been introduced in the area of train operation systems. The work in [18; 31; 32; 33] attempted to solve the train automatic stop control problem using machine learning methods. Specifically, [PERSON] et al. [33] employed a regression neural network to estimate the train station parking error. The algorithm parameters are deliberately adjusted to reduce parking errors. In [32], the authors proposed adopting regression algorithms to analyze the train stopping accuracy. By extracting relevant features through feature engineering, the ridge regression and the elastic net regression are employed to build models to reflect the relationship between the features and the stopping accuracy. These studies only consider the train stop control problem that occurs when the train stops at stations. In [17], coordinated iterative learning control schemes were designed for train trajectory tracking. This method can learn to improve control performances from previous executions, which requires less system model knowledge. The speed tracking errors are reduced via repeated operations of the train. Three data-driven train operation models were proposed in [19]. By summarizing an expert system from the domain driving experience, K-nearest neighbor, Bagging-CART and Adaboost are employed for train operation. Then, these models are improved via a heuristic train parking algorithm to ensure the parking accuracy. Compared to the methods in [32; 33], the whole operation process is analyzed in this paper. Reinforcement learning based algorithms were employed to solve optimal control problems in [34]. An adaptive actor-critic learning framework is carefully designed with the virtual reference feedback tuning. This approach was validated to learn nonlinear state-feedback control for linear output reference model tracking, which is innovative and can be applied to the train control system. Besides, data-driven methods also have been used for fault detection and diagnosis in the area of rail transit [35]. These inspiring studies give us a new angle to view the problem of building the train dynamic model. Data-driven algorithms have been widely used in intelligent transportation systems [16; 36; 37].