Fatigue load extrapolation is an important research topic in modern mechanical engineering. It plays a key role in predicting the service life of industrial equipment. In real-world applications, long-term load testing is often costly and time-consuming. Therefore, researchers aim to estimate full-life load histories using short-term data through various models and algorithms [1]-[3].

Traditional statistical methods have been widely used for load spectrum extrapolation. Common techniques include kernel density estimation (KDE), Weibull distribution, and extreme value theory (EVT). Yang et al. [4] improved KDE by using the DBSCAN algorithm to optimize bandwidth, which helped capture the tail distribution of load more accurately. Shen et al. [5] developed a dual-spectrum extrapolation method that considered the sequence effects of loads in drilling systems. Yong Zhang et al. [6] analyzed metro vehicle loads and proposed a programmatic method to generate load spectra. Mei et al. [7] combined multibody dynamics with statistical modeling to improve pantograph load prediction. These traditional models are easy to apply and interpret [8]. However, they often fail to capture rare extreme loads that are critical for fatigue damage.

In recent years, deep learning and data-driven models have become powerful tools for load extrapolation and fatigue life prediction. Heng et al. [9] proposed a CNN-LSTM hybrid model to predict multiaxial fatigue life across different materials. Chen et al. [10] developed a frequency-domain neural network that improved the prediction of fatigue over multiple scales. Santos et al. [11] introduced a physics-guided neural network to estimate offshore wind turbine loads more accurately. Farid [12] combined artificial neural networks with Gaussian process regression to enable real-time fatigue prediction under random loading. Jia et al. [13] used an LSTM model to predict the fatigue life of reinforced concrete beams, showing strong learning ability even with complex materials. These studies show that deep learning models can better capture nonlinear relationships in fatigue data than traditional statistical methods. Other researchers have focused on data reconstruction and hybrid modeling approaches. Lei et al. [14] used generative adversarial networks (GANs) to reconstruct missing load data in structural health monitoring. Gibson et al. [15] applied Gaussian process regression to predict strain and fatigue damage distributions. Yang et al. [16] and Zheng et al. [17] used extreme value theory and the peaks-over-threshold method to capture rare load cycles more accurately. Dewa and Kepka [18] improved extrapolation accuracy for bus structural steel, and Yang et al. [19] proposed a hybrid time-domain extrapolation method for vehicle loads. These studies suggest that combining engineering knowledge with data-driven modeling can significantly improve load extrapolation accuracy.

Recent studies have also worked on improving the design of loss functions and applying statistical constraints in neural networks. Ustinova and Lempitsky [20] proposed the histogram loss to align the model’s output distribution with the target data. Niu et al. [21] developed a tail-aware prediction framework using LSTM and the generalized extreme value (GEV) distribution to better predict rare events. Imani et al. [22] tested histogram loss in regression tasks and found that it improved both model stability and convergence. These methods help neural networks maintain consistency with the statistical characteristics of real fatigue data while avoiding overfitting. Some recent studies have also introduced advanced hybrid frameworks and application-specific deep neural networks. Shi et al. [23] built a load spectrum model for loader equipment to support fatigue life evaluation. Gan et al. [24] created a modular neural network, pretrained with uniaxial fatigue data, to predict multiaxial fatigue life. Wang et al. [25] proposed a multi-physics neural network for additively manufactured superalloy parts, showing strong generalization ability across different materials. These works demonstrate that combining engineering knowledge, data-driven modeling, and deep learning enables high-accuracy and robust fatigue life predictions.

Despite the noticeable achievements in load spectrum compilation and extrapolation, current rainflow-based and statistical methods still face two key limitations: (1) Most parametric and KDE-based models focus on fitting the overall range-mean distribution, but they lack explicit mechanisms to control or reproduce extreme load cycles. (2) Many extrapolation strategies aim to match amplitude-level statistics; however, the generated time histories are typically assembled from sampled cycles without enforcing local smoothness or temporal consistency, making them less realistic in the time domain.

To overcome these challenges, this study proposes a Tail-Weighted Histogram-Regularized Long Short-Term Memory network (TWHR-LSTM).

(1) The method introduces a tail-weighted sampling strategy combined with penalties on large step changes between adjacent points, allowing the model to pay more attention to high-amplitude segments and preserve them more accurately in the extrapolated results.

(2) A sequence-to-sequence LSTM framework is trained with pointwise reconstruction and variance regularization to capture temporal patterns, while a differentiable amplitude-histogram loss ensures that the generated distribution aligns with the original.

Together, these design elements improve both the fidelity of extreme events and the time-domain realism of the extrapolated load sequence.

In this work, a tail-weighted histogram-regularized long short-term memory network (TWHR-LSTM) is proposed to extrapolate a 10-second seed load into a 100 s load sequence. The raw force signal is first detrended to remove linear drift and then normalized. The resulting sequence is split chronologically into training and validation sets. Input-output pairs are created using overlapping sliding windows along the 10-second input, and a smooth tail-weighted sampling strategy is applied. This increases the sampling frequency for segments with large peak-to-peak ranges, while still retaining cycles of moderate amplitude.

The model adopts a sequence-to-sequence LSTM with an encoder-decoder architecture and a monotonic calibration layer. It is trained using a composite loss function that includes pointwise reconstruction error, a differentiable amplitude-histogram distance, variance regularization, and a tail penalty based on large local increments to reflect severe rainflow cycles. After training, the model operates in autoregressive mode, and short overlapping output blocks are combined using cross-fading to generate a continuous 100 s load sequence (see Figure 1).

Figure 1. Training TWHR-LSTM model and load extrapolation.

Experimental data used in this study were collected from field tests on the powertrain mounting system of a heavy-duty vehicle. The powertrain was elastically connected to the vehicle body at four mounting points (see Figure 2). Triaxial accelerometers were installed at each mount to record vibrations in three directions at a sampling rate of 2000 Hz. All signals were synchronously recorded using a dynamic acquisition system.

Figure 2. Heavy-duty vehicle powertrain mounting system.

Engineering experience indicated that the left-rear mount is more prone to fatigue damage during operation. Therefore, this study focuses on the dynamic force signal in the z-direction at that location. The test was conducted under a steady vehicle speed of 30 km/h, and the total recording time was 100 seconds.

The raw signal included high-frequency noise, so a low-pass filter with a 500 Hz cutoff was applied to improve the signal-to-noise ratio while preserving relevant content. For load spectrum analysis, rainflow counting was used to remove cycles that contribute little to fatigue. Specifically, cycles with amplitudes below 10% of the maximum load were discarded. A full 100-second load-time history was then reconstructed using linear interpolation (see Figure 3).

Figure 3. 100 s force data at 30 km/h.

To evaluate different extrapolation methods, a validation framework was designed. Starting from the first 10 seconds of measured data, each method generated an extrapolated 100 s sequence. These results were compared with the original measured 100 s signal for performance assessment.

It is important to acknowledge that the proposed method is condition-specific. Since the spectral characteristics and amplitude distributions of powertrain vibrations vary significantly with vehicle speed, the model requires retraining when applied to a different operating condition (e.g., 60 km/h). However, given the lightweight architecture of the TWHR-LSTM, retraining on a new 10-second seed is computationally efficient, allowing for rapid adaptation to various steady-state scenarios.

This section presents three extrapolation strategies that extend a 10 s load signal to 100 s. All share the same detrended 10 s seed and produce signals with identical sampling rate and target duration. The three approaches tested reflect different philosophies of modeling. Two classical rainflow methods model cycle statistics and render synthetic turning points as a means of indirect time series reconstruction. The proposed TWHR-LSTM learns continuation directly in the time domain, with explicit attention to tail enhancement and distribution matching.

This section compares three extrapolation strategies. The comparison focuses on distribution-oriented metrics: the rainflow from-to matrix, the range exceedance curve, and the pseudo-damage ratio with respect to the measured 100 s record. These together reflect whether or not an extrapolated signal captures both small-cycle behavior and the tail-dominated cycles which dominate fatigue.

This paper investigated data-driven extrapolation of road load spectra from a short 10 s seed load to an equivalent 100 s sequence. Two conventional rainflow-based approaches were used as baselines: one parametric method using Weibull-distributed ranges, Gaussian-distributed means, and a Gaussian copula; and one non-parametric KDE-based method. Both methods can reproduce the overall range histogram, but they are sensitive to tail fitting and copula assumptions.

A tail-weighted histogram-regularized LSTM (TWHR-LSTM) was then proposed. The network is trained in a sequence-to-sequence setting with sliding windows, using tail-aware sampling, a differentiable histogram loss to enforce amplitude consistency, and a penalty on large step changes to preserve rare, high-amplitude cycles.

Rainflow from-to matrices and range exceedance curves show that TWHR-LSTM better preserves the shape of the tail region. Pseudo-damage per unit time confirms this trend: the proposed method yields a pseudo-damage ratio closest to the real 100 s reference. Overall, the results indicate that TWHR-LSTM can provide a more balanced load extrapolation, capturing extreme events without sacrificing global statistical consistency.