Differentiable Fatigue Damage Modeling for Neural-Network-Based Virtual Sensing

Abstract

Virtual sensors implemented in vehicles enable the prediction of forces experienced during their operational lifetime and thus allow an extension of durability assessment into the operational phase of customer vehicles.

Due to limited computational capacity of onboard electronic control units, full-scale deep learning models are often too expensive for real-time operation. As a result, compact neural network architectures are employed. However, their limited model capacity prevents them from capturing all relevant characteristics of the underlying load history, leading to significant deviations in fatigue damage estimation based on
Rainflow Counting compared to the true damage.

To address this limitation, this thesis proposes a fatigue damage surrogate for Rainflow Counting based on the spectral method by Dirlik. The underlying spectral estimates, in particular the periodogram and the Welch method, are investigated with respect to their influence on the resulting damage, their computational efficiency, and their differentiability. Ensuring differentiability down to the spectral estimation level enables the integration of the damage surrogate as an additional loss term within the deep learning training process.

It is shown that the physics-informed Dirlik formulation produces extremely small gradients, introducing a bottleneck in the training process due to vanishing-gradient-like behaviour. By adapting the loss function, this issue is mitigated through effective gradient rescaling, allowing stable optimization.

The results demonstrate that training with a multitask-like loss improves the prediction of fatigue damage during inference. These findings provide a framework for enhancing virtual sensor predictions with respect to fatigue damage assessment.

Popular Abstract

Fatigue damage is caused by repeated stress that vehicle components experience throughout their operational lifetime. For example, when a car drives over uneven road surfaces such as bumps or potholes, small deformations occur in certain components. Over time, these repeated stresses accumulate and lead to material damage.

To monitor this process and improve vehicle design, virtual sensors are used to estimate the loads acting on a vehicle during real operation. These sensors are based on machine learning models, which are trained to predict the loads in real-time by minimizing the difference between measured (true) load and model predictions. During training, the model iteratively updates its parameters to improve this prediction.

However, in practical applications these models must run directly inside the vehicle, where computational resources are limited. Therefore, only relatively small neural network architectures can be used. Such models often fail to capture extreme events, for example large stress peaks caused by driving over deep potholes. Although these events occur less frequently, they have a significant impact on fatigue damage.

Fatigue damage is commonly computed using so-called counting methods, which identify and evaluate peaks and valleys in the load-time history. When extreme values are incorrectly predicted, these methods produce inaccurate damage estimates. To address this issue, fatigue damage is incorporated directly into the training objective of the model.

Since traditional counting methods are not differentiable and therefore cannot be used within gradient-based optimization, this thesis replaces them with a spectral fatigue damage model proposed by Dirlik. Spectral methods are based on the power spectral density of a signal and, as shown in this work, can be formulated in a differentiable way. By ensuring differentiability of the entire method, including the spectral estimation step, the damage can be integrated directly into the training process.

However, the gradients resulting from the Dirlik formulation are extremely small, which can prevent effective learning. This issue is addressed by adapting the loss function, allowing these gradients to be rescaled and enabling stable training.

The results show that incorporating fatigue damage into training improves the model’s ability to predict damage. In particular, the model becomes better at capturing extreme events, reducing the mismatch between predicted and actual damage. A remaining challenge is that different load histories can result in similar damage values, which may lead to variability in training results and requires further investigation.

• author: Conrad, Katharina
• supervisor: Mengwu Guo; David Neuhäuser; Ruben Schwiedernoch
• organization: Mathematical Statistics; Mathematics (Faculty of Sciences); Centre for Mathematical Sciences; Mathematics (Faculty of Engineering)
• alternative title: Modellering av differentierbar utmattningsskada för virtuell avkänning baserad på neurala nätverk
• course: NUMM03 20261
• year: 2026
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Numerical analysis, virtual sensor, fatigue damage, differentiable modeling, neural network, machine learning, optimization, vanishing gradient
• publication/series: Master’s Theses in Mathematical Sciences
• report number: LUNFNA-3049-2026
• ISSN: 1404-6342
• other publication id: 2026:E79
• language: English
• additional info: Project Framework: This research was carried out within the framework of a joint project with Mercedes-Benz AG. The thesis was officially evaluated and approved by both the university faculty and the designated corporate supervisors.
• id: 9244498
• date added to LUP: 2026-07-13 14:55:31
• date last changed: 2026-07-13 14:55:31