# Temporal Convolutional Network-Based Axle Load Estimation from pavement Vibration Data
## 1 Introduction  
Capturing vehicle axle weight is crucial in weight enforcement and assessing pavement condition. Accurate axle load estimates assist in regulatory compliance, targeted inspections, load-induced pavement damage assessment, pavement design and planning, and optimized maintenance scheduling [1; 2; 3]. Weigh in motion (WIM) is an efficient method for measuring axle weight while a vehicle is moving at normal speed. Traditional WIM systems use load cells, bending plates, and piezoelectric sensors [4; 5; 6; 7] to measure the pressure between tires and the road. However, these sensors are not only expensive but also impractical when vehicles drive too fast, as there is insufficient time to register changes in pressure. Furthermore, these sensors must be installed on the road surface, and the connection between sensors and road structures deteriorates under repeated traffic loads, leading to increasing measurement errors [8]. Vibration-based weigh-in-motion methods have gained popularity in recent years, as traffic-induced pavement vibration contains valuable information on traffic loads and pavement structure. The measured vibration data can be utilized for traffic monitoring [9; 10]and structural health monitoring [11; 12] through vibration analysis. [PERSON] et al. [13] developed acceleration sensing nodes to measure pavement vibration and identified abnormal axle weights based on the collected vibration data. [PERSON] et al. [14] developed a cost-effective MEMS accelerometer to measure traffic-induced vibration and estimate the axle weights of trucks. Vibration sensors have a sampling frequency of up to 512 Hz, which is more appropriate than pressure sensors at high speeds. However, there are challenges associated with vibration-based WIM methods. Firstly, pavement vibrations attenuate when transmitted over a distance due to damping [15; 16], leading to low signal-to-noise ratio (SNR) measurements when traffic loads are too far away from the vibration sensor. Moreover, conventional vibration sensors are single-point sensors with limited spatial measurement range, making it challenging to ensure that moving tires are located exactly on those sensors [17; 18]. Secondly, pavement vibrations are related to pavement conditions, requiring the removal of structural effects during vibration analysis. The process is based on vibration characteristics in the time and frequency domains, such as frequency response functions, modal coordinates [19; 20; 21], convolution integrals, and transfer functions [22; 23; 24], which typically require complicated mathematical modeling and formula derivation. Thus, analyzed results are susceptible to measurement noise and mathematical model errors [25; 26]. Distributed optical vibration sensing (DOVS) technology provides a promising solution to overcome these challenges. DOVS-based vibration sensing systems can measure traffic-induced vibration with a high sampling frequency and precise location accuracy. [PERSON] et al. [27] developed a DOVS-based vibration sensing system to estimate the category of vehicles and load level through time-frequency analysis. [PERSON] et al. [28; 29; 30; 31] utilized DOVS to reconstruct vehicle-induced vibration and analyze the modal characteristics of concrete pavement to recognize damage. [PERSON] et al. [32; 33] proposed a data processing method for this system, which included denoising and cutting vibration periods. The processed data are utilized to estimate the speed, position, and category of vehicles. For vibration analysis, recurrent neural networks (RNNs) are well suited for investigating the complex nonlinear relationship between input and output data through a training process [34; 35; 36; 37]. Compared to traditional RNNs, temporal convolutional networks (TCN) have equivariance properties and sparse interactions that translate well into the time domain. The infinite memory advantage of RNNs is largely absent in practice. TCN exhibit longer memory than recurrent architectures with the same capacity. A thorough comparison by [PERSON] et al.