**Ngoan T. Do, Mustafa Gül \* and Saeideh Fallah Nafari**

Department of Civil & Environmental Engineering, University of Alberta, Edmonton, AB T6G 1H9, Canada; tndo@ualberta.ca (N.T.D.); fallahna@ualberta.ca (S.F.N.)

**\*** Correspondence: mustafa.gul@ualberta.ca; Tel.: +1-780-492-3002

Received: 12 May 2020; Accepted: 18 June 2020; Published: 22 June 2020

**Abstract:** Track foundation stiffness (also referred as the track modulus) is one of the main parameters that affect the track performance, and thus, quantifying its magnitudes and variations along the track is widely accepted as a method for evaluating the track condition. In recent decades, the train-mounted vertical track deflection measurement system developed at the University of Nebraska–Lincoln (known as the MRail system) appears as a promising tool to assess track structures over long distances. Numerical methods with different levels of complexity have been proposed to simulate the MRail deflection measurements. These simulations facilitated the investigation and quantification of the relationship between the vertical deflections and the track modulus. In our previous study, finite element models (FEMs) with a stochastically varying track modulus were used for the simulation of the deflection measurements, and the relationships between the statistical properties of the track modulus and deflections were quantified over different track section lengths using curve-fitting approaches. The shortcoming is that decreasing the track section length resulted in a lower accuracy of estimations. In this study, the datasets from the same FEMs are used for the investigations, and the relationship between the measured deflection and track modulus averages and standard deviations are quantified using artificial neural networks (ANNs). Different approaches available for training the ANNs using FEM datasets are discussed. It is shown that the estimation accuracy can be significantly increased by using ANNs, especially when the estimations of track modulus and its variations are required over short track section lengths, ANNs result in more accurate estimations compared to the use of equations from curve-fitting approaches. Results also show that ANNs are effective for the estimations of track modulus even when the noisy datasets are used for training the ANNs.

**Keywords:** railroad tracks; track modulus; computer simulation; artificial neural networks
