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Physics-Informed Machine Learning—An Emerging Trend in Tribology

Department of Mechanical and Metallurgical Engineering, School of Engineering, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul 6904411, Chile
Engineering Design and CAD, University of Bayreuth, Universitätsstr. 30, 95447 Bayreuth, Germany
Author to whom correspondence should be addressed.
Lubricants 2023, 11(11), 463;
Submission received: 6 September 2023 / Revised: 24 September 2023 / Accepted: 2 October 2023 / Published: 30 October 2023
(This article belongs to the Special Issue Recent Advances in Machine Learning in Tribology)


Physics-informed machine learning (PIML) has gained significant attention in various scientific fields and is now emerging in the area of tribology. By integrating physics-based knowledge into machine learning models, PIML offers a powerful tool for understanding and optimizing phenomena related to friction, wear, and lubrication. Traditional machine learning approaches often rely solely on data-driven techniques, lacking the incorporation of fundamental physics. However, PIML approaches, for example, Physics-Informed Neural Networks (PINNs), leverage the known physical laws and equations to guide the learning process, leading to more accurate, interpretable and transferable models. PIML can be applied to various tribological tasks, such as the prediction of lubrication conditions in hydrodynamic contacts or the prediction of wear or damages in tribo-technical systems. This review primarily aims to introduce and highlight some of the recent advances of employing PIML in tribological research, thus providing a foundation and inspiration for researchers and R&D engineers in the search of artificial intelligence (AI) and machine learning (ML) approaches and strategies for their respective problems and challenges. Furthermore, we consider this review to be of interest for data scientists and AI/ML experts seeking potential areas of applications for their novel and cutting-edge approaches and methods.

Graphical Abstract

1. Artificial Intelligence and Machine Learning in Tribology

The complex interactions between surfaces in relative motion or between surfaces and flowing media have substantial impacts on the performance, efficiency, and service life of tribo-technical systems. In recent years, the integration of artificial intelligence (AI) and machine learning (ML) techniques in tribology has opened up new possibilities for improving understanding, prediction, and control of friction, lubrication, and wear phenomena [1,2]. AI refers to the development of intelligent machines that are capable of performing tasks that typically require human intelligence. ML is a subfield of AI (see Figure 1) and focuses on the development of experience-based algorithms that allow for computers to learn and make predictions or decisions (output) based on data (input) without being explicitly programmed [3]. Some notable ML techniques encompass decision trees (tree-like structures that make decisions based on feature values) [3], random forests (combining multiple decision trees to improve prediction accuracy) [4], support vector machines (aiming to find the best decision boundary between classes in a dataset) [5], and neural networks, just to mention a few. Among these techniques, artificial neural networks (ANNs) have gained significant prominence. They consist of interconnected “neurons”, organized into layers, whereby each neuron receives an input, performs computations, and passes the result to the next layer. Through training, i.e., adjusting the connections’ weights and biases, complex patterns in the data can be captured [3,6,7].
All of these ML/AI approaches possess the potential to revolutionize tribology by enabling more accurate modeling, efficient optimization, and an enhanced control of friction and wear processes [1]. One of the primary applications of AI and ML in tribology is predictive modeling by analyzing large datasets, thus identifying patterns and hidden relationships that may not be apparent through traditional analytical methods [9,10,11,12]. Moreover, AI and ML techniques can facilitate condition-based maintenance and real-time monitoring in tribological systems when employing respective integrated sensors and data acquisition systems [13,14,15]. Furthermore, AI and ML can contribute to designing and optimizing tribo-systems within vast design spaces [16] or can even contribute to discovering novel solutions that may not have been considered previously. All of these aspects may lead to the development of more efficient lubricants [17,18] and materials [19,20], advanced surface modifications [21,22], manufacturing processes [23,24], or innovative tribo-system designs [25,26], not only going beyond mere buzzwords, but actually resulting in improved energy efficiency, reduced emissions, and an enhanced overall system performance [27].
Meanwhile, there is a number of review articles showcasing the usages and many promises of AI and ML within tribology [1,2,28,29,30,31]. However, a challenge remains in the training of AI/ML models, which relies heavily on the availability of large amounts of high-quality experimentally [32,33,34,35,36,37,38] or numerically [39,40,41,42] generated data. Ideally, these data should be FAIR (Findable, Accessible, Interoperable, and Reusable), meaning it should be well documented, easily accessible, compatible with different systems, and suitable for reuse in different contexts [43,44,45]. However, acquiring such data for scientific or industrial tribology problems can often be challenging, and these data may not always be readily available [46,47]. Also, relying on data alone bears the risks of having misunderstood the scientific problem and not converging towards generalizability.
As an alternative to data-based AI strategies, in situations where there is a scarcity of available data, ML models can be trained using supplementary data derived from the application of physical laws, incorporating mathematical models. This approach, known as physics-informed ML (PIML), thus connects the big data regime, without any knowledge about the underlying physics, with the area of small data and lots of physics [48] (see Figure 2). The employment of PIML in tribology is likewise a comparatively new as well as emerging trend, which has not been covered by other review articles yet. This article therefore seeks to shed some light on the novel trend of physics-informed ML. The concept will be briefly introduced in Section 2, the current state of the art will be discussed in detail in Section 3, and the article will end with some concluding remarks in Section 4.

2. Physics-Informed Machine Learning

PIML is an approach that combines ML techniques with the principles and constraints of physics to enhance the accuracy, interpretability, and generalizability of models [48,49]. PIML aims to address the sole dependence on data by incorporating prior knowledge of physics into the learning process, ensuring that the resulting models align with the fundamental principles of the domain [48]. Thus, PIML models can capture the underlying physics, even in cases where the available data are limited, noisy, or incomplete. This integration allows for models that are not only data-driven, but also consistent with the fundamental principles governing the system [48]. By incorporating physics-based knowledge, it is possible to enhance the predictive accuracy compared to conventional, data-driven ML approaches. Furthermore, physics-informed models are often more interpretable, which allows for a better understanding of the underlying mechanisms and optimizing tasks. Thereby, physics-informed models, once properly trained with a solid understanding of the physics involved, can be adapted to various applications and environments with relatively minor adjustments. Finally, by incorporating physical laws, machine learning models are less likely to make predictions that violate fundamental principles, reducing the risk of erroneous or unrealistic results, e.g., predicting a negative film thickness in hydrodynamic contacts, etc. Apart from the observational biases contained in a sufficiently large dataset, as used to train classical ML models, it may consist of inductive biases through a direct intervention into the ML model architecture, for example, in the form of mathematical constraints to be strictly satisfied that are known a priori [48]. Furthermore, learning biases can be incorporated into the training phase through the careful selection of loss functions, constraints, and inference algorithms [48]. These can effectively guide the model towards converging on solutions that align with the fundamental principles of physics [48]. By incorporating soft penalty constraints and fine-tuning them, it becomes possible to approximately satisfy the underlying physical laws, offering a flexible framework to introduce a wide range of physics-based biases, expressed through integrals or differential equations [48]. Observational, inductive, or learning biases are not mutually exclusive and can be combined synergistically to create a diverse set of hybrid approaches to construct PIML systems [48].
Even though a variety of approaches are generally available [50], the most common methodology in PIML is the use of Physics-Informed Neural Networks (PINNs), which combine artificial neural networks with physics-based equations, such as differential equations or conservation laws [49,51]. During the training phase, these equations are incorporated into the loss functions of a neural network to guide the learning process, i.e., there is a data-driven part and a physics-driven part in the loss function. The neural network learns to approximate both the data-driven aspects and the physics-based constraints simultaneously, resulting in models that capture the complex interactions between data and physics [49]. As illustrated in Figure 3, this is achieved by sampling a set of input training data (i.e., spatial coordinates and/or time stamps) and passing it through the neural network. Subsequently, the network’s output gradients are computed with respect to its inputs at these locations. These gradients can frequently be analytically obtained via auto-differentiation (AD) and are then used to calculate the residual of the underlying differential equation. The residual is then incorporated as an additional term in the loss function. The aim of including this “physics loss” in the loss function is to guarantee that the solution learned by the network aligns with the established laws of physics.
Another approach in PIML involves the utilization of probabilistic models, such as Gaussian processes or Bayesian inference, to incorporate physical priors and uncertainties into the learning process [48]. These models enable the quantification of uncertainty and the propagation of physical constraints through the machine learning framework [48].
The applications of PIML are wide-ranging and can be found in various scientific and engineering domains. It has been employed in fluid dynamics for flow prediction and turbulence modeling [52,53,54], in material science to predict material behavior [55,56,57,58] and discover new materials [59], in structural mechanics [60,61], medical imaging [62,63], and many other fields where physical laws play crucial roles. By integrating physics-based knowledge into machine learning models, PIML also offers a powerful tool for understanding and optimizing tribological phenomena and thus represents a very recent and emerging trend in the domain of tribology.

3. Physics-Informed Machine Learning in Tribology

3.1. Lubrication Prediction

PIML can be applied to various tribological tasks, for example, the prediction of lubrication conditions and the optimization of lubrication processes. By considering the governing equations of fluid dynamics and incorporating experimental or simulation data, ML models can learn to predict the lubricant film thickness, pressure, and/or shear stress distribution. As such, Almqvist [64] implemented a PINN in MATHWORKS Matlab to solve the Reynolds boundary value problem (BVP) in a linear slider, assuming a one-dimensional flow of an incompressible and iso-viscous fluid. The rather simple feedforward neural network consisted of one input node (coordinate x), one hidden layer (i.e., a single layer network) with ten neurons, as well as one output node (see Figure 4a), and employed the sigmoid activation function. The Reynolds BVP was described by a second-order ordinary differential equation:
x H 3 p x = H x , f o r   0 < x < 1
with the dimensionless film thickness H(x) and the dimensionless pressure p(x). The pressure at the boundaries was chosen to be zero (p(0) = 0, p(1) = 0). The Reynolds BVP was then condensed to
H 3 p + H 3 p H = 0   f o r   0 < x < 1   ,
p ( 0 ) p ( 0 ) = 0   ,
and the loss function was defined as
L = H x 3 p + H x 3 p H x 2 + p 2 0 + p 2 1
After establishing the partial derivatives of p’’ and p(1) with respect to the weights and bias instead of the commonly employed AD, Almqvist [64] used the PINN approach to solve for the dimensionless pressure in a linear slider with a converging gap of the form H(x) = 2 − x and compared the result to an exact analytical solution (see Figure 4b). Thereby, an overall error of 6.2 × 10−5 as well as errors of 4.1 × 10−4 at x = 0 and −4.0 × 10−4 at x = 1 were obtained. It is worth noting that this approach does not offer advantages neither with respect to accuracy nor efficiency compared to the established finite difference (FDM) or finite element method-based solutions, but it presents a meshless approach, and not a data-driven approach [64], thus overcoming the “curse of dimension” [65]. Furthermore, cavitation effects were not considered by this formulation, and the study was limited to solving the one-dimensional Reynolds equation for the pressure at a given film thickness profile.
Inspired by the pioneering work from Almqvist [64], several authors have taken up the idea and extended the PINN approach. As such, Zhao et al. [66] solved for the two-dimensional Reynolds equation:
x H 3 p x = L B y H 3 p y 6 H x
for a slider bearing with the length L and width B as well as zero-pressure conditions at the edges. The film thickness was described as
H x = θ L h 0 1 x + 1
with the inclination of the slider θ and the outlet film thickness h0. The PINN was programmed in Julia language and followed the examples of [49,67]. The authors studied the influence of the number of training epochs (i.e., the number of complete iterations through the model training process, where the model learns from the available physics-based knowledge, constraints, or equations, making incremental adjustments to its parameters in an effort to improve its performance) as well as the influences of the layer and neuron numbers on the predicted pressure distribution. They reported that the maximum values converged fairly well, while the pressure at the boundaries of the domain as well as the global loss took some more epochs (see Figure 5a). Furthermore, Zhao et al. [66] compared different PINN topologies without hidden layers, with one hidden layer, as well as with two hidden layers with 16 neurons each. As depicted in Figure 5b, while the pressures in the central region were somewhat comparable, the PINN without hidden layers displayed strongly fluctuating pressures at the edges; thus, it strongly diverged from the zero-pressure boundary conditions. In turn, the differences between the PINNs with one hidden layer and two hidden layers were neglectable. Similarly, using fewer neurons in the hidden layers (e.g., four) led to undesired pressure fluctuations at the boundary of the domain, while using either 16 or 32 nodes did not affect the results in a significant way (see Figure 5c). The authors concluded that a PINN topology with 16 neurons in one hidden layer as well as 1000 training epochs allow for a satisfactory solution of the Reynolds equation.
Moreover, Zhao et al. [66] integrated the PINN into an iterative solution process (Figure 6a) for the pressure and film thickness distribution, thus balancing an externally applied load W:
Ω p x , y d x d y = W h 0 2 η U L 2 B   ,
whereby η is the lubricant viscosity and u the sliding velocity. Zhao et al. further verified this developed iterative PINN approach against the results obtained using the finite element method (FEM) as well as the experimentally measured values obtained by means of optical interferometry in a slider-on-disk setup (see Figure 6b). Generally, an excellent agreement was observed. Even though the pressure at the boundaries did not strictly meet the zero-pressure condition in the case of the PINN (deviations up to 3.4%), an excellent correlation between the PINN and FEM prevailed in the majority of the domain (Figure 6c), which was manifested in an overall error of 1.5% between the two.
Li et al. [68] employed a PINN to solve the Reynolds equation to predict the pressure field and film thickness of a gas-lubricated journal bearing (assuming incompressibility) in order to subsequently calculate the aerodynamic characteristics under variable eccentricity ratio conditions (see Figure 7a,b). The authors compared the results with an FDM solution and reported that the PINN could capture the flow field structure quite well (Figure 7c,d). Thereby, the convergence accuracy was reported to be improved by changing the weight values of different loss items as well as by employing a second-order optimizer to fine-tune the results. Moreover, the authors performed a comprehensive comparison (Figure 7e,f) among three different learning strategies (unsupervised and supervised learning driven by data from FDM, semi-supervised learning with sufficient data, and semi-supervised learning with a small number of noisy data) with respect to the prediction accuracy, i.e., the difference between the predicted results and true physics, and the physics interpretability, which describes the degree to which the results meet the physical equations. It was observed that the data-driven supervised learning method had the best prediction accuracy without a sharp loss increase in the boundary cases, followed by semi-supervised learning, and finally, unsupervised learning. In turn, the supervised learning method did not meet the Reynolds equation and had no interpretability, while the unsupervised and semi-supervised methods satisfied the physics conservation equation with small losses. However, the accuracy of the semi-supervised approach tended to be reduced with noisier data, but not the interpretability. Li et al. [68] concluded that the learning method generally should be chosen based upon the prediction accuracy requirement for the actual application as well as the amount of available data. In situations where there is a lack of experimental or high-precision numerical solution data, the unsupervised learning approach offers a direct solution to approximate the prediction value of the flow field. Thus, it becomes possible to obtain an estimation without relying on specific data or prior knowledge. However, when there is a limited amount of data available, the semi-supervised learning method can be employed to achieve more accurate prediction outcomes. This considers both solution accuracy and physics interpretability, leading to improved results and eliminating the need for simulations in each individual case, which is typically required by conventional numerical methods. In contrast, when complete field physics values are directly provided, the data-driven method can accurately predict the flow field for unknown conditions without possessing physical interpretability.
Yadav and Thakre [69] also employed a PINN to study the behavior of a fluid-lubricated journal as well as a two-lobe bearing and compared the obtained results against an FEM model. Even though the authors provided few insights and details on the employed model and its implementation, they reported a quite good correlation between the PINN and FEM at various load cases, with errors below 6% and 5% with respect to the predicted eccentricity and friction coefficient.
Xi et al. [70] investigated the application of PINNs to predict the pressure distribution of a finite journal bearing and compared the results when employing soft or hard constraints for the boundary conditions (see Figure 8a and Figure 8b, respectively). The models were implemented in the Python library, DeepXDE, whereby the ANN consisted of three hidden layers with 20 neurons each, and tanh was used as the activation function. The PINN was trained to minimize the loss function using the Gradient Descent Method, and the Adam optimizer was used to obtain the weights. The Dirichlet boundary condition was employed for the Reynolds equation in the case of the soft constraint (Figure 8a). Furthermore, the authors converted the boundary condition into a hard one (Figure 8b) by modifying the neural network, in which the boundary condition could be satisfied. Also, the boundary condition was no longer part of the loss function. Thus, the hard constraint met the pressure boundary condition in a mathematically exact manner and sped up the convergence. The authors compared the developed approaches as well as the FDM results when assuming both constant and variable (temperature-dependent) viscosity, whereby a good agreement was reported.
In the aforementioned studies, the cavitation effects were neglected since they reduce the complexity. Rom [71] extended the idea of using PINNs for lubrication prediction towards the consideration of cavitation by introducing the fractional film content θ to the Reynolds equation,
x H 3 p x + y H 3 p y = 6 η u   H θ x
which was solved with the following underlying constraints:
p 0 ,   0 θ 1 ,   p 1 θ = 0 .
This means that the computational domain was split into two sub-domains, i.e., the full film region with the conventional Reynolds equation (p > 0, θ = 1) and the cavitated region (p = 0, θ < 1). A priori, the boundary in between the two regions is free and unknown, which makes it complex for conventional algorithms. In turn, strictly dividing both domains is not necessary for PINNs when covered by suitable boundary conditions. Rom [71] specified these problem-/application-specific conditions for the example of journal bearings (see Figure 8a). The author first employed a residual neural network (ResNet) (see Figure 8b), and training was conducted to minimize the error with respect to the mentioned boundary conditions as well as the residual (Reynolds equation divided by H), which was derived via AD. Moreover, the approach was extended to not only develop a PINN for one specific problem (fixed set of parameters), but to account for variable parameters; in this case, the variable eccentricity was the parameter, which was also propagated as the input parameter through the ResNet (extended PINN) (see Figure 8c). This led to a certain generalizability of the model. The loss function consisted of three losses related to the predictions of p and θ on the boundaries as well as three global losses. The neural network parameters were initialized via Glorot initialization and then optimized using a limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm. Tanh was chosen as the activation function and for the output layer. While the fractional film content was between zero and one, this required scaling of the input variables as well as re-scaling of the pressure with an arbitrary chosen upper boundary to obtain dimensional results. Since abrupt jumps and the fractional film content can complicate training, Rom [71] proposed to adaptively add collocation points during the training, i.e., refining the region around the maximum pressure and the boundary between the pressure and cavitation region. The author compared the obtained results for the standard and extended PINNs against the FDM solutions and found a pretty good agreement (see Figure 8d–g). Using a total of 20 neurons in six hidden layers proved to achieve the best results. The errors in between the prediction for the maximum pressure, load carrying capacity, and frictional force at different eccentricities were below 1.6%, 0.3%, and 0.2%, respectively, thus verifying certain generalizability (Figure 8g). However, some minor differences were observed, especially at the transition from the pressure to cavitated region (Figure 8d,e), which were attributed to the high resolution of the FDM region, while the PINN encountered difficulties with the jump in the fractional film content.
To overcome the manual or computationally expensive initial value threshold selection as well as the weight adjustment/optimization of Rom’s approach, Cheng et al. [72] very recently presented a PINN framework for computing the flow field of hydrodynamic lubrication by solving the Reynolds equation while involving cavitation effects by means of the Swift–Stieber model [73,74] as well as the Jakobsson–Floberg–Olsson (JFO) [75,76] model. The authors introduced a penalizing scheme with a residual of non-negativity and an imposing scheme with a continuous differentiable non-negative function to satisfy the non-negativity constraint of the Swift–Stieber approach. To address the complementarity constraint inherent to the JFO theory, the pressure and cavitation fractions were considered as the outputs of the neural network, and the Fischer–Burmeister (FB) equation’s residual enforced their complementary relationship. Chen et al. then employed multi-task learning (MTL) techniques (dynamic weight, uncertainty weight, and projecting conflicting gradient method) to strike a balance between optimizing the functions and satisfying the constraints. This was shown to be superior to traditional penalizing schemes. To finally assess the accuracy of their approach, the authors studied the setup of an oil-lubricated 3D journal bearing at a fixed eccentricity with Dirichlet boundary conditions, showing very low errors compared to the respective FEM models.

3.2. Wear and Damage Prediction

Apart from predicting the lubrication phenomena in hydrodynamically or aerodynamically lubricated contacts, PIML has been employed for wear prediction. Haviez et al. [77] suggested the use of a semi-physical neural network when addressing fretting wear and facing scarce datasets due to testing costs and efforts, thus overcoming the drawbacks of purely data-driven ML. To this end, the authors experimentally generated 53 datasets using a fretting wear tester. The two-step semi-PINN was trained without backpropagation or any regularization method simply by introducing (approximate) physical considerations about energy dissipation,
E d μ = α 0 N α 1 δ α 2 F α 3
and asperity contact to estimate the wear volume,
V = α   E d μ β
according to Archard’s law, whereby μ is the coefficient of friction, N is the number of fretting cycles, δ is the sliding amplitude, F is the normal force, and α, α1, α2, α3, α, and β are the fitting parameters to be adjusted according to the input–output relations obtained from the experiments (see Figure 9). Following linearization by taking the logarithmic approach, a single-layer ANN with an exponential activation function and a simple least squares approximation were used to determine the unknown parameters. Despite its simplicity, the authors reported a good generalizability of the suggested approach in terms of the relative quadratic error (RQE) on the new testing data, outperforming conventional ANNs when trained with small data, which might feature overfitting. Yet, it should be considered that fitting an ANN to rather simple analytical functions might be an unnecessary complication compared to other regression methods.
Yucesan and Viana [78] suggested a hybrid PIML approach consisting of a recurrent neural network to develop a cumulative damage model to predict the fatigue of wind turbine main bearings. Thereby, the physics-informed layers were used to model the comparatively well-understood physics, i.e., the bearing lifetime, while a data-driven layer accounted for the aspects, which have so far been beyond the scope of physical modeling, i.e., grease degradation (see Figure 10). The reason was because the input conditions, such as the loads and temperatures, are fully observed over the entire time series, while grease conditions are typically only partially observed at distinct inspection intervals. The model takes the bearing fatigue damage increment.
d t B R G = n t 1 60 N i t i a 1 a S K F C P 10 3 ,
where the number of passed cycles is nt, the total operational hours is ti, the velocity is Ni, the basic dynamic load rating is C, the equivalent dynamic bearing load is P, and the reliability and life modification factors are a1 and aSKF. In contrast, the grease damage increment ∆dtGRS, i.e., the degradation of viscosity and increasing contamination, was implemented via a multilayer perceptron. The recurrent neural network then took the wind speed WSt (mapped to equivalent bearing loads) and bearing temperature T as inputs, thus updating the respective parameters and calculating the cumulative wear. The authors employed their approach to several load cases from real wind turbine data (10 min average operational and monthly grease inspection data for 14 turbines) and demonstrated that the general trends regarding bearing damage and grease degradation could be covered fairly well. Thereby, it was shown that the selection of the initialization of the weights of multilayer perceptron is crucial, and that a set of initial weights that is far away from optimum would not lead to accurate predictions. However, this can be improved by “engineering judgement-based weight initialization” [78], i.e., by performing a sensitivity analysis on the general influence trends of the inputs, thus selecting favorable initial weights.
Similarly, Shen et al. [79] proposed an approach for bearing fault detection that integrates principles of physics with deep learning methodologies. The approach consisted of two integral components: a straightforward threshold model and a convolutional neural network (CNN). The threshold model initiated the assessment of bearing health statuses by applying established physics principles associated with bearing faults. By following this initial evaluation, the CNN autonomously extracted significant high-level features from the input data, effectively utilizing these features to predict the bearing’s health class. To facilitate the incorporation of physics-based knowledge into the deep learning model, the authors developed a loss function that selectively enhanced the influence of the physics-based insights assimilated by the threshold model when embedding this knowledge into the CNN model. To validate the efficacy of their approach, Shen et al. conducted experiments using two distinct datasets. The first dataset comprised data collected from 18 bearings operating in the field of an agricultural machine, while the second dataset contained data from bearings subjected to testing in the laboratory at the Case Western Reserve University (CWRU) Bearing Data Center.
Ni et al. [80] recently presented a physics-informed framework for rolling bearing diagnostics, whereby data were collected from a test rig under varying operating conditions, such as different speeds and loads. The primary difficulties were extracting robust physical information under these diverse conditions and integrating it into the network’s architecture. To this end, a first layer was created using cepstrum exponential filtering, emphasizing the modal properties in the signal. The modal properties, being linked to the system characteristics rather than specific operating conditions, offered robustness to varying conditions. The layer served to establish a network that can operate effectively across diverse operating scenarios, including transitions from healthy to faulty states or changes in fault locations. Another layer based on computed order tracking (COT) converted time domain signals into angle domain signals, removing the influence of rotational speed variations and allowing for the extraction of distinctive bearing fault features under conditions of variable or time-varying speeds. Following the initial layers, a parallel bi-channel Physics-Informed Residual Network (PIResNet) architecture was implemented. The processing in the one channel was initiated with the domain conversion layer, followed by the inclusion of a wide kernel CNN layer for the purpose of mitigating high-frequency noise. Subsequently, two residual building blocks (RBBs) and max pooling layers were sequentially introduced. In contrast, the other channel commenced with a modal-property-dominant-generated layer aimed at enhancing the modal properties that were closely tied to the intrinsic characteristics of the system, making them less susceptible to changes in the operating conditions. The remainder of this channel mirrored the configuration of the other with the objective of automatically extracting complex high-dimensional features from the modal-property-dominant signal. Upon completing their respective processes, both channels were flattened and combined. Following this fusion, the fully connected and softmax layers were used for the purpose of classification. The effectiveness of this approach was verified through experiments involving bearings operating under varying speeds, loads, and time-varying speed conditions. Comprehensive comparisons confirmed the excellent performance of the PIResNet in terms of high accuracy, adaptability to different load and speed scenarios, and resilience to noise.
Li et al. [81] presented a PIML framework to predict machining tool wear under varying tool wear rates, consisting of the three modules of piecewise fitting, a hybrid physics-informed data-driven model, and automatic learning (meta-learning) (see Figure 11a). Initially, a piecewise fitting strategy was adopted to estimate the empirical equation parameters and to calculate the tool wear rate in initial, normal, and severe wear states. Subsequently, the physics-informed data-driven (PIDD) model inputs were determined using the parameters derived from the piecewise fitting approach. Utilizing a cross physics–data fusion strategy, i.e., fusing the data and the physical domain, these inputs, along with the local features, were then mapped to the tool wear rate space, thus creating the physics-informed model. Finally, meta-learning was employed to acquire an understanding of the dependable correlations between the tool wear rate and force throughout the tool’s lifespan. To enhance interpretability and maintain the physical consistency of the PIML model, a physics-informed loss term was formulated, which served to improve the interpretability of the meta-learning process while ensuring that the PIML model adhered to the governing fundamental physical principles. The authors compared the developed approach for multiple sensory data (vibration, acoustic emission, etc.) and the tool flank wear observations from conducted cutting experiments with various deep learning and conventional machine learning models. Thereby, the proposed PIML framework could relatively accurately predict the tool wear trends and featured a substantially higher accuracy than a bi-directional backward gated recurrent unit (Bi-GRU) neural network, a CNN, long short-term memory (LSTM), and support vector regression (SVR) (see Figure 11b).

4. Concluding Remarks

To sum up, PIML has gained significant attention in various scientific fields and is now emerging in the area of tribology. By integrating physics-based knowledge into ML, PIML offers potential for understanding and optimizing tribological phenomena, overcoming the drawbacks of traditional ML approaches that rely solely on data-driven techniques. As discussed within Section 3 and summarized in Table 1, PIML can be applied to various tribological tasks.
As such, PINNs have been employed for lubrication prediction by solving the Reynolds differential equation. Starting with the 1D Reynolds equation for a converging slider, in only two years, the complexity has already been tremendously increased, now covering the 2D Reynolds equation, journal bearings with load balance and variable eccentricity, and cavitation effects. A common limitation of PINNs is that a low loss in terms of the residual of the partial differential equation does not necessarily indicate a small prediction error. Therefore, in the future, it will be crucial to gain experience with these novel techniques to find the most effective algorithms, configurations, and hyperparameters. Future work should also be directed towards expanding the PINN’s capabilities by replacing the Reynolds equation with formulations that consider nonstationary flow behavior, lubricant compressibility, or shear-thinning fluids, thus addressing a wider range of application scenarios and obtaining more accurate solutions in various lubrication contexts. Moreover, further input parameters should be incorporated into the Reynolds or film thickness equation. After training, which undoubtedly would be more complex and time-consuming, this would ultimately allow for extensive parameter studies to be conducted for optimization tasks, e.g., of textured surfaces [82], and facilitate faster computation, making it promising for solving elastohydrodynamic problems where the pressure and film thickness need to be computed repeatedly in an iterative procedure until convergence is achieved [71]. Thereby, the computational efficiency and overall accuracy might further be improved by parallel neural networks and extreme learning machines [83,84] as well as advanced adaptive methods, e.g., residual point sampling [85].
With regard to wear and damage prediction, semi or hybrid PIML approaches have been employed so far, combining empirical laws and equations with experimentally obtained data. Since testing costs and efforts are generally high or data are simply scarce, these approaches tend to feature advantages compared to purely data-driven ML methods in terms of the prediction accuracy. Since wear processes are inherently strongly statistical and underly scatter, future work might incorporate the Bayesian approach within PIML for uncertainty consideration and quantification. Thereby, a prior distribution is augmented over the model parameters, representing the initial belief about their values. By combining this prior distribution with the observed data, a posterior distribution is obtained, representing the updated beliefs about the parameters given the data. This would ultimately favor the handling of limited and noisy data as well as the ability to quantify uncertainty, providing valuable insights into the reliability of predictions. Furthermore, models used with the aim of predicting damage in real-word tribo-technical systems have so far mainly focused on rolling bearings. Future research should seek to explore the applicability of PIML to other mechanical systems like gears. Such investigations could broaden the scope of the employed method’s use towards vibration-based gear and surface wear propagation monitoring.

Author Contributions

Conceptualization, M.M. and S.T.; methodology, formal analysis, and writing—original draft preparation, M.M.; writing—review and editing, S.T.; visualization, M.M. All authors have read and agreed to the published version of the manuscript.


This work was partially supported by the European Regional Development Fund in Bavaria under the Gate2HPC project.

Data Availability Statement

Not applicable.


M.M. greatly acknowledges the financial support from the Vicerrectoría Académica (VRA) of the Pontificia Universidad Católica de Chile within the Programa de Inserción Académica (PIA). S.T. kindly acknowledges the continuous support of the University of Bayreuth, Germany.

Conflicts of Interest

The authors declare no conflict of interest.


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Figure 1. Classification of the terms artificial intelligence, machine learning, deep learning, data mining, and physics-informed machine learning. Redrawn and adapted from [8].
Figure 1. Classification of the terms artificial intelligence, machine learning, deep learning, data mining, and physics-informed machine learning. Redrawn and adapted from [8].
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Figure 2. Data and physics scenarios. Redrawn and adapted from [48].
Figure 2. Data and physics scenarios. Redrawn and adapted from [48].
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Figure 3. Graphical representation of a PINN approach.
Figure 3. Graphical representation of a PINN approach.
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Figure 4. (a) Topology of the employed PINN to solve Reynolds BVP and (b) comparison of the PINN prediction for a linear converging slider with the exact solution. Reprinted and adapted from [64] with permission from CC BY 4.0.
Figure 4. (a) Topology of the employed PINN to solve Reynolds BVP and (b) comparison of the PINN prediction for a linear converging slider with the exact solution. Reprinted and adapted from [64] with permission from CC BY 4.0.
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Figure 5. Pressure distribution (a) after 100 training epochs (left), 500 (middle) and 1000 (right) training epochs, (b) after training without hidden layers (left), with one hidden layer (middle), and with two (right) hidden layers (16 neurons each) as well as (c) after training with 4 (left), 16 (middle), and 32 neurons in one hidden layer. Reprinted and adapted from [66] with permission from CC BY 4.0.
Figure 5. Pressure distribution (a) after 100 training epochs (left), 500 (middle) and 1000 (right) training epochs, (b) after training without hidden layers (left), with one hidden layer (middle), and with two (right) hidden layers (16 neurons each) as well as (c) after training with 4 (left), 16 (middle), and 32 neurons in one hidden layer. Reprinted and adapted from [66] with permission from CC BY 4.0.
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Figure 6. (a) Flow chart of the iterative PINN approach for hydrodynamic contact. (b) Outlet film thickness at different sliding velocities for the PINN method compared to FEM simulation as well as experimental results. (c) Pressure distribution predicted using the PINN (left) and the FEM (right). Reprinted and adapted from [66] with permission from CC BY 4.0.
Figure 6. (a) Flow chart of the iterative PINN approach for hydrodynamic contact. (b) Outlet film thickness at different sliding velocities for the PINN method compared to FEM simulation as well as experimental results. (c) Pressure distribution predicted using the PINN (left) and the FEM (right). Reprinted and adapted from [66] with permission from CC BY 4.0.
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Figure 7. (a) Structure of a gas-lubricated journal bearing. (b) PINN topology to solve the Reynolds equation. (c) Comparison of flow field and (d) aerodynamic characteristics between PINN (prediction) and FDM (true). (e) Loss function curves against testing data as well as (f) L2 loss comparison for pressure and film thickness at different eccentricities for semi-supervised, unsupervised, and supervised learning methods. Reprinted and adapted from [68] with permission.
Figure 7. (a) Structure of a gas-lubricated journal bearing. (b) PINN topology to solve the Reynolds equation. (c) Comparison of flow field and (d) aerodynamic characteristics between PINN (prediction) and FDM (true). (e) Loss function curves against testing data as well as (f) L2 loss comparison for pressure and film thickness at different eccentricities for semi-supervised, unsupervised, and supervised learning methods. Reprinted and adapted from [68] with permission.
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Figure 8. (a) Cartesian domain for a journal bearing with respective boundary conditions. (b) Standard and (c) extended PINN architecture used to solve the Reynolds equation with respective boundary conditions to consider cavitation. Comparison and error between extended PINN and FDM with respect to the (d) pressure and (e) the fractional film content. Pressure and fractional film content along the contact length for (f) the training values of the eccentricity and (g) eccentricity values not employed for training. Reprinted and adapted from [71] with permission.
Figure 8. (a) Cartesian domain for a journal bearing with respective boundary conditions. (b) Standard and (c) extended PINN architecture used to solve the Reynolds equation with respective boundary conditions to consider cavitation. Comparison and error between extended PINN and FDM with respect to the (d) pressure and (e) the fractional film content. Pressure and fractional film content along the contact length for (f) the training values of the eccentricity and (g) eccentricity values not employed for training. Reprinted and adapted from [71] with permission.
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Figure 9. Semi-PINN two-level structure used to predict fretting wear. Reprinted from [31] with permission from CC BY 4.0.
Figure 9. Semi-PINN two-level structure used to predict fretting wear. Reprinted from [31] with permission from CC BY 4.0.
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Figure 10. Hybrid PINN for main bearing fatigue and grease degradation. Reprinted from [78] with permission from CC BY 3.0.
Figure 10. Hybrid PINN for main bearing fatigue and grease degradation. Reprinted from [78] with permission from CC BY 3.0.
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Figure 11. (a) Proposed PIML framework and (b) predicted tool wear in x-direction of the proposed model compared with various ML approaches. Reprinted and adapted from [81] with permission.
Figure 11. (a) Proposed PIML framework and (b) predicted tool wear in x-direction of the proposed model compared with various ML approaches. Reprinted and adapted from [81] with permission.
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Table 1. Overview of PIML approaches reported in the literature with their fields of application.
Table 1. Overview of PIML approaches reported in the literature with their fields of application.
Field of ApplicationPIML ApproachYearReference
Lubrication predictionUsing PINN to solve the 1D Reynolds BVP to predict the pressure distribution in a fluid-lubricated linear converging slider2021[64]
Using PINN to solve the 2D Reynolds equation to predict the pressure and film thickness distribution considering load balance in a fluid-lubricated linear converging slider2023[66]
Using supervised, semi-supervised, and unsupervised PINN to solve the 2D Reynolds equation to predict the pressure and film thickness distribution considering load balance and eccentricity in a gas-lubricated journal bearing2022[68]
Using PINN to solve the 2D Reynolds equation to predict the behavior of fluid-lubricated journal as well as two-lobe bearings2023[69]
Using PINN with soft and hard constraints to solve the 2D Reynolds equation to predict the pressure distribution in fluid-lubricated journal bearings at fixed eccentricity with constant and variable viscosity2023[70]
Using PINN to solve the 2D Reynolds equation to predict the pressure and fractional film content distribution in fluid-lubricated journal bearings at fixed and variable eccentricity considering cavitation2023[71]
Using PINN to solve the 2D Reynolds equation to predict the pressure and fractional film content distribution in fluid-lubricated journal bearings at fixed eccentricity considering cavitation2023[72]
Wear and damage predictionUsing semi PINN to find regression fitting parameters for Archard’s wear law based upon small data from fretting wear experiments2015[77]
Using hybrid PINN to predict wind turbine bearing fatigue based upon a physics-informed bearing damage model as well as data-driven grease degradation approach2020[78]
Using physics-informed CNN with preceding threshold model for rolling bearing fault detection2021[79]
Using physics-informed residual network for rolling bearing fault detection2023[80]
Using PIML framework consisting of piecewise fitting, a hybrid physics-informed data-driven model, and meta-learning to predict tool wear2022[81]
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Marian, M.; Tremmel, S. Physics-Informed Machine Learning—An Emerging Trend in Tribology. Lubricants 2023, 11, 463.

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