⚡ electrical engineering

Vibrations forecasting using physics informed machine learning: case-study of self-lubricated hydrodynamic journal bearings

This study presents a hybrid physics-informed machine learning framework that effectively forecasts vibration states in self-lubricated hydrodynamic journal bearings by integrating approximate physical models with Bayesian parameter correction, achieving high accuracy and generalization even with limited training data and incomplete physical knowledge.

Original authors: S. Shevchik, S. Jalikop, T Ivas, J. Prost, G. Vorlaufer, M. Varga, P. Hoffmann

Published 2026-10-14

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CC BY 4.0

Original authors: S. Shevchik, S. Jalikop, T Ivas, J. Prost, G. Vorlaufer, M. Varga, P. Hoffmann

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Machines that spin, slide, and grind are the silent workhorses of modern industry, from the tiny bearings in a car's alternator to the massive turbines powering a city. For decades, engineers have relied on physics to predict how these parts wear down, using equations that describe the forces of friction, heat, and pressure. But real machines are messy. They operate in environments that change, with parts that degrade in unpredictable ways, making the perfect equations of the past often fall short of the reality of today. When a machine fails, it is often too late; the damage is done, and the cost is high. The challenge for scientists has long been to bridge the gap between the clean laws of physics and the chaotic, data-rich reality of a machine that is slowly breaking.

A team of researchers has developed a new way to look at this problem, combining the reliability of physical laws with the adaptability of modern computer learning. Instead of trying to build a single, perfect model of a machine or relying solely on vast amounts of historical data, they created a hybrid system that learns as the machine ages. By testing this approach on a specific type of bearing used in windscreen wipers and valves, they demonstrated that it is possible to forecast how a machine will vibrate hours into the future, even when the machine is changing in ways the original engineers never anticipated. This method offers a path to predicting failures before they happen, using far less data than traditional artificial intelligence requires.

The researchers focused their study on a self-lubricating journal bearing, a component where a shaft spins inside a sleeve lined with a material that holds oil like a sponge. These bearings are designed to run smoothly for a long time, but eventually, the friction and heat cause the materials to wear down, changing how the machine moves. To understand this process, the team built a test rig that mimics the start-stop conditions these bearings face in the real world. They placed the bearings on a pendulum-like setup and subjected them to repeated cycles of speed changes, heating them up to simulate years of use in a fraction of the time. As the bearings wore, the team measured the vibrations and the torque, or twisting force, generated by the spinning shaft.

The core of their innovation lies in how they used these measurements. Instead of feeding raw vibration data directly into a computer program to guess the future, they first used the data to update a simplified physical model of the machine. Think of this model as a set of rules describing how the pendulum should move based on gravity, friction, and the stiffness of the materials. In the beginning, the model was a good guess, but as the bearing wore down, the real machine began to behave differently than the model predicted. The researchers used a statistical method to constantly tweak the numbers inside the model, adjusting them to match the real-world vibrations they were measuring. This allowed the model to stay accurate even as the machine changed.

Once the model was updated to reflect the current state of the machine, the team used a specialized type of artificial intelligence to look ahead. This AI, known as a neural operator, did not try to predict the exact vibration signal for every second of the future. Instead, it learned how the numbers inside the physical model would evolve over time. By predicting how the model's parameters would change over the next eight hours, the system could then run the physical model forward to generate a forecast of what the vibrations would look like. This approach is powerful because it keeps the physical meaning of the machine intact while using the learning power of AI to handle the complexity of wear and tear.

The results of this experiment were striking. When the researchers tested their system on data from bearings that had not yet failed, their method predicted future vibrations with an error rate of less than 25% for horizons extending more than eight hours. This was significantly better than other machine learning methods that tried to predict the vibrations directly without using the physical model as a guide. Those other methods struggled, often producing errors that were nearly double or triple the size of the new approach. The hybrid system was also able to generalize, meaning it worked well even when the machine was operating under conditions that were slightly different from the ones it had seen during training.

Perhaps the most telling test came when the researchers applied their system to a bearing that eventually failed. They trained the system only on data from healthy bearings, never showing it what a broken machine looked like. When the bearing in the test began to fail, the system's predictions started to drift away from the reality of the vibrations. However, this drift was not a failure of the system; it was a signal. The gap between what the system expected and what was actually happening grew large enough to serve as an early warning that something was wrong. The system successfully captured the overall behavior of the machine even as it broke down, suggesting that this approach could be used to detect failures that have never been seen before.

This work demonstrates that the future of machine monitoring may not lie in choosing between physics and data, but in weaving them together. By letting a physical model evolve with the machine and using AI to forecast that evolution, the researchers have created a tool that is both robust and adaptable. It requires less data than pure machine learning and offers more insight than pure physics. For industries that rely on the continuous operation of complex machinery, this could mean the difference between a planned maintenance stop and a catastrophic breakdown, turning the silent wear of metal into a readable story of the machine's health.

Technical Summary: Vibrations Forecasting Using Physics-Informed Machine Learning

1. Problem Statement

The study addresses the challenge of forecasting future vibration states in complex engineering systems, specifically self-lubricated hydrodynamic journal bearings (PJBs), under conditions of limited training data, incomplete physical knowledge, and a lack of failure examples.

Traditional condition monitoring often relies on detecting anomalies after they occur, which may leave insufficient time for corrective action. Furthermore, standard Machine Learning (ML) approaches, including Physics-Informed Neural Networks (PINNs), face significant hurdles in industrial predictive maintenance:

Data Scarcity: Fault and failure data are difficult to reproduce and collect, limiting the generalization of data-driven models.
Model Uncertainty: Governing physical equations often require boundary conditions or correction terms that are difficult to determine experimentally or are unknown due to wear, degradation, and changing component interactions.
Static Assumptions: Conventional models often assume fixed parameters, failing to capture the non-stationary dynamics of degrading mechanical assemblies.

The authors propose a hybrid framework designed to forecast vibration dynamics over extended horizons (up to 8 hours) by combining physical knowledge with data-driven adaptation, specifically targeting systems where a complete physical description is infeasible.

2. Methodology

The proposed framework is a hybrid Physics-Informed Machine Learning (PIML) architecture that operates in three distinct stages: parameter inference, parameter forecasting, and signal reconstruction.

2.1. Modeling Strategies

The study employs two parallel modeling approaches to handle varying levels of physical knowledge:

Physics-Based Model: An idealized two-degree-of-freedom ordinary differential equation (ODE) model representing the test rig as a pendulum. It incorporates viscous torque, gravitational restoring torque, damping, and rod deformation. The model is non-dimensionalized using four key parameters ( $\Pi_e, \Pi_v, \Pi_g, \Pi_d$ ) representing elasticity, viscosity, gravity, and damping effects relative to inertia.
Physics-Surrogate Model: A data-driven ODE model constructed using the PySINDy library (Sparse Identification of Nonlinear Dynamical Systems). This model represents system dynamics as a sparse combination of candidate functions (polynomial terms involving torque, vibration, and shaft speed) when accurate physics-based models are unavailable or too expensive to develop.

2.2. Bayesian Parameter Inference

To address the evolution of system dynamics (e.g., wear and lubricant degradation), the framework updates model parameters for each test run using Bayesian inference.

Process: The measured torque and shaft speed are used as inputs to the models. The model parameters (either the non-dimensional numbers for the physics model or the weights $\mathbf{W}$ for the surrogate model) are treated as unknown variables.
Optimization: Two optimization strategies were investigated:
- Classical Bayesian Optimization: Uses a Gaussian Process surrogate and an Expected Improvement (EI) acquisition function to minimize the squared error between modeled and measured vibrations.
- Deep Bayesian Optimization: Uses deep learning to approximate posterior distributions, suitable for highly nonlinear problems, though it required significantly more computational iterations.
Outcome: This step yields a probability distribution over the parameter space, with the mean used to update the model for the current state of the machine.

2.3. Neural Operator Forecasting

Instead of forecasting vibration signals directly, the framework forecasts the temporal evolution of the model parameters.

Architecture: Neural Operators (NOs) are employed to learn mappings between functional spaces. Unlike standard neural networks that map finite-dimensional inputs to outputs, NOs learn mappings between infinite-dimensional function spaces.
Function: The NO takes the current updated parameters and forecasts their evolution over the prediction horizon (the interval between test runs).
Reconstruction: The predicted future parameters are fed back into the respective dynamical model (physics or surrogate) to reconstruct the future vibration response. This approach preserves physical interpretability while leveraging the generalization capabilities of NOs for nonlinear systems.

2.4. Experimental Setup

The framework was validated using data from a Multiple Bearing Wear Tribometer (MSVT) at AC2T research GmbH.

Specimen: Porous journal bearings (bronze and iron-based) pre-impregnated with lubricant.
Conditions: Accelerated lifetime tests under a constant radial load (250 N) with a bidirectional sawtooth speed profile (-20 to +20 rpm).
Data Collection: Vibration was measured via laser displacement sensors. Stribeck ramps (0 to 3000 rpm) were periodically executed to capture dynamic responses.
Dataset: Six experimental campaigns were conducted. The framework was trained on one non-failure campaign and evaluated on four other non-failure campaigns and one failure campaign.

3. Key Contributions

Hybrid PIML Framework for Non-Stationary Systems: The paper introduces a framework that combines approximate physical models with data-driven parameter correction (via Bayesian inference) to track evolving machine parameters without extensive retraining.
Parameter-Space Forecasting: Unlike methods that predict raw signals, this approach uses Neural Operators to forecast the evolution of model parameters. This reduces the forecasting problem to a low-dimensional space, improving robustness and interpretability.
Surrogate Model Integration: The study demonstrates the utility of physics-surrogate models (based on SINDy) as computationally efficient alternatives to complex physics models, particularly when underlying physics is uncertain.
Zero-Shot Failure Forecasting: The framework was tested on a failure campaign without any failure data in the training set, demonstrating its ability to detect deviations from normal operation and forecast dynamics under unseen failure conditions.

4. Results

The framework was evaluated against conventional ML models (Fully Connected NN, CNN, ResNet) and standard PINN formulations.

Forecasting Accuracy: The proposed framework achieved forecasting errors below 25% for horizons exceeding 8 hours. Specifically, the physics-surrogate model combined with Neural Operators achieved a relative mean difference of 16% ± 8%, outperforming all reference models (which ranged from 24% to 54% error).
Generalization: The framework consistently outperformed reference models and generalized to operating conditions not represented in the training data.
Failure Detection: In the failure case (trained only on non-failure data), the framework maintained accuracy during early stages. Upon failure onset (marked by a sudden vibration increase), error rates increased to approximately 26–28%, primarily due to amplitude mismatches. However, the system successfully captured the overall vibration dynamics and the onset of the anomaly.
Computational Efficiency: Down-sampling signals to 1,000 points preserved prediction accuracy while significantly reducing computational time. Classical Bayesian optimization required ~8 iterations, whereas Deep Bayesian optimization required ~100 iterations.

5. Significance and Claims

The authors claim that this work demonstrates the potential of hybrid PIML for predictive maintenance in complex engineering systems where physical descriptions are unavailable, incomplete, or difficult to reconstruct.

Data Efficiency: By forecasting in the parameter space rather than the signal space, the framework reduces the dependency on large, diverse training datasets, making it suitable for industrial applications where fault data is scarce.
Adaptability: The use of Bayesian inference allows the system to adapt to non-stationary dynamics (wear, degradation) in real-time without the operational downtime associated with retraining conventional ML models.
Interpretability: The approach retains the physical interpretability of the underlying models while leveraging the flexibility of machine learning.
Applicability: The generic architecture is presented as applicable to systems with uncertain physics, offering a viable alternative to pure data-driven approaches in scenarios with limited data availability.

The paper concludes that while larger datasets might favor deep learning approaches, the proposed framework offers a robust solution for the specific constraints of industrial condition monitoring, balancing accuracy, interpretability, and data efficiency.