# Physics-Informed Machine Learning—An Emerging Trend in Tribology

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Artificial Intelligence and Machine Learning in Tribology

## 2. Physics-Informed Machine Learning

## 3. Physics-Informed Machine Learning in Tribology

#### 3.1. Lubrication Prediction

^{−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.

_{0}. 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.

#### 3.2. Wear and Damage Prediction

_{1}, α

_{2}, α

_{3}, 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.

_{t}, the total operational hours is t

_{i}, the velocity is N

_{i}, the basic dynamic load rating is C, the equivalent dynamic bearing load is P, and the reliability and life modification factors are a

_{1}and a

_{SKF}. In contrast, the grease damage increment ∆d

_{t}

^{GRS}, i.e., the degradation of viscosity and increasing contamination, was implemented via a multilayer perceptron. The recurrent neural network then took the wind speed WS

_{t}(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.

## 4. Concluding Remarks

**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].

**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

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Marian, M.; Tremmel, S. Current Trends and Applications of Machine Learning in Tribology—A Review. Lubricants
**2021**, 9, 86. [Google Scholar] [CrossRef] - Rosenkranz, A.; Marian, M.; Profito, F.J.; Aragon, N.; Shah, R. The Use of Artificial Intelligence in Tribology—A Perspective. Lubricants
**2021**, 9, 2. [Google Scholar] [CrossRef] - Bell, J. Machine Learning: Hands-On for Developers and Technical Professionals; Wiley: Hoboken, NJ, USA, 2014; ISBN 978-1-118-88906-0. [Google Scholar]
- Breiman, L. Random Forests. Mach. Learn.
**2001**, 45, 5–32. [Google Scholar] [CrossRef] - Schölkopf, B.; Smola, A.J. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond; MIT Press: Cambridge, MA, USA, 2002; ISBN 978-0-262-19475-4. [Google Scholar]
- Sarkar, D.; Bali, R.; Sharma, T. Practical Machine Learning with Python: A Problem-Solver’s Guide to Building Real-World Intelligent Systems; Apress: Berkeley, CA, USA, 2017; ISBN 978-1-4842-3206-4. [Google Scholar]
- Kruse, R.; Borgelt, C.; Braune, C.; Klawonn, F.; Moewes, C.; Steinbrecher, M. Computational Intelligence: Eine Methodische Einführung in Künstliche Neuronale Netze, Evolutionäre Algorithmen, Fuzzy-Systeme und Bayes-Netze, 2nd ed.; Überarbeitete und Erweiterte Auflage; Springer Vieweg: Wiesbaden, Germany, 2015; ISBN 978-3-658-10904-2. [Google Scholar]
- Gerschütz, B.; Sauer, C.; Wallisch, A.; Mehlstäubl, J.; Kormann, A.; Schleich, B.; Alber-Laukant, B.; Paetzold, K.; Rieg, F.; Wartzack, S. Towards Customized Digital Engineering: Challenges and potentials of adapting digital engineering methods for the product development process. In Stuttgarter Symposium für Produktentwicklung SSP 2021; Fraunhofer IAO, Ed.; Fraunhofer IAO: Stuttgart, Germany, 2021; pp. 93–104. [Google Scholar]
- Kurt, H.I.; Oduncuoglu, M. Application of a Neural Network Model for Prediction of Wear Properties of Ultrahigh Molecular Weight Polyethylene Composites. Int. J. Polym. Sci.
**2015**, 2015, 315710. [Google Scholar] [CrossRef] - Vinoth, A.; Datta, S. Design of the ultrahigh molecular weight polyethylene composites with multiple nanoparticles: An artificial intelligence approach. J. Compos. Mater.
**2020**, 54, 179–192. [Google Scholar] [CrossRef] - Hasan, M.S.; Kordijazi, A.; Rohatgi, P.K.; Nosonovsky, M. Triboinformatics Approach for Friction and Wear Prediction of Al-Graphite Composites Using Machine Learning Methods. J. Tribol. Trans. ASME
**2022**, 144, 011701. [Google Scholar] [CrossRef] - Hasan, M.S.; Kordijazi, A.; Rohatgi, P.K.; Nosonovsky, M. Triboinformatic modeling of dry friction and wear of aluminum base alloys using machine learning algorithms. Tribol. Int.
**2021**, 161, 107065. [Google Scholar] [CrossRef] - Kanai, R.A.; Desavale, R.G.; Chavan, S.P. Experimental-Based Fault Diagnosis of Rolling Bearings Using Artificial Neural Network. J. Tribol. Trans. ASME
**2016**, 138, 031103. [Google Scholar] [CrossRef] - Prost, J.; Cihak-Bayr, U.; Neacșu, I.A.; Grundtner, R.; Pirker, F.; Vorlaufer, G. Semi-Supervised Classification of the State of Operation in Self-Lubricating Journal Bearings Using a Random Forest Classifier. Lubricants
**2021**, 9, 50. [Google Scholar] [CrossRef] - Argatov, I.; Jin, X. Time-delay neural network modeling of the running-in wear process. Tribol. Int.
**2023**, 178, 108021. [Google Scholar] [CrossRef] - Marian, M.; Grützmacher, P.; Rosenkranz, A.; Tremmel, S.; Mücklich, F.; Wartzack, S. Designing surface textures for EHL point-contacts—Transient 3D simulations, meta-modeling and experimental validation. Tribol. Int.
**2019**, 137, 152–163. [Google Scholar] [CrossRef] - Dai, K.; Gao, X. Estimating antiwear properties of lubricant additives using a quantitative structure tribo-ability relationship model with back propagation neural network. Wear
**2013**, 306, 242–247. [Google Scholar] [CrossRef] - Bhaumik, S.; Pathak, S.D.; Dey, S.; Datta, S. Artificial intelligence based design of multiple friction modifiers dispersed castor oil and evaluating its tribological properties. Tribol. Int.
**2019**, 140, 105813. [Google Scholar] [CrossRef] - Padhi, P.K.; Satapathy, A. Analysis of Sliding Wear Characteristics of BFS Filled Composites Using an Experimental Design Approach Integrated with ANN. Tribol. Trans.
**2013**, 56, 789–796. [Google Scholar] [CrossRef] - Gangwar, S.; Pathak, V.K. Dry sliding wear characteristics evaluation and prediction of vacuum casted marble dust (MD) reinforced ZA-27 alloy composites using hybrid improved bat algorithm and ANN. Mater. Today Commun.
**2020**, 25, 101615. [Google Scholar] [CrossRef] - Sahraoui, T.; Guessasma, S.; Fenineche, N.E.; Montavon, G.; Coddet, C. Friction and wear behaviour prediction of HVOF coatings and electroplated hard chromium using neural computation. Mater. Lett.
**2004**, 58, 654–660. [Google Scholar] [CrossRef] - Boidi, G.; Rodrigues da Silva, M.; Profito, F.J.J.; Machado, I.F. Using Machine Learning Radial Basis Function (RBF) Method for Predicting Lubricated Friction on Textured and Porous Surfaces. Surf. Topogr. Metrol. Prop.
**2020**, 8, 044002. [Google Scholar] [CrossRef] - Gupta, S.K.; Pandey, K.N.; Kumar, R. Artificial intelligence-based modelling and multi-objective optimization of friction stir welding of dissimilar AA5083-O and AA6063-T6 aluminium alloys. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl.
**2018**, 232, 333–342. [Google Scholar] [CrossRef] - Anand, K.; Shrivastava, R.; Tamilmannan, K.; Sathiya, P. A Comparative Study of Artificial Neural Network and Response Surface Methodology for Optimization of Friction Welding of Incoloy 800 H. Acta Metall. Sin. (Engl. Lett.)
**2015**, 28, 892–902. [Google Scholar] [CrossRef] - Francisco, A.; Lavie, T.; Fatu, A.; Villechaise, B. Metamodel-Assisted Optimization of Connecting Rod Big-End Bearings. J. Tribol. Trans. ASME
**2013**, 135, 041704. [Google Scholar] [CrossRef] - Zavos, A.; Katsaros, K.P.; Nikolakopoulos, P.G. Optimum Selection of Coated Piston Rings and Thrust Bearings in Mixed Lubrication for Different Lubricants Using Machine Learning. Coatings
**2022**, 12, 704. [Google Scholar] [CrossRef] - Tremmel, S.; Marian, M. Machine Learning in Tribology—More than Buzzwords? Lubricants
**2022**, 10, 68. [Google Scholar] [CrossRef] - Paturi, U.M.R.; Palakurthy, S.T.; Reddy, N.S. The Role of Machine Learning in Tribology: A Systematic Review. Arch Comput. Methods Eng
**2023**, 30, 1345–1397. [Google Scholar] [CrossRef] - Sose, A.T.; Joshi, S.Y.; Kunche, L.K.; Wang, F.; Deshmukh, S.A. A review of recent advances and applications of machine learning in tribology. Phys. Chem. Chem. Phys.
**2023**, 25, 4408–4443. [Google Scholar] [CrossRef] - Yin, N.; Xing, Z.; He, K.; Zhang, Z. Tribo-informatics approaches in tribology research: A review. Friction
**2023**, 11, 1–22. [Google Scholar] [CrossRef] - Argatov, I. Artificial Neural Networks (ANNs) as a Novel Modeling Technique in Tribology. Front. Mech. Eng.
**2019**, 5, 1074. [Google Scholar] [CrossRef] - Boidi, G.; Grützmacher, P.G.; Varga, M.; Da Rodrigues Silva, M.; Gachot, C.; Dini, D.; Profito, F.J.; Machado, I.F. Tribological Performance of Random Sinter Pores vs. Deterministic Laser Surface Textures: An Experimental and Machine Learning Approach. In Tribology of Machine Elements-Fundamentals and Applications; IntechOpen: London, UK, 2021. [Google Scholar] [CrossRef]
- de La Guerra Ochoa, E.; Otero, J.E.; Tanarro, E.C.; Morgado, P.L.; Lantada, A.D.; Munoz-Guijosa, J.M.; Sanz, J.M. Optimising lubricated friction coefficient by surface texturing. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2013**, 227, 2610–2619. [Google Scholar] [CrossRef] - Gyurova, L.A.; Miniño-Justel, P.; Schlarb, A.K. Modeling the sliding wear and friction properties of polyphenylene sulfide composites using artificial neural networks. Wear
**2010**, 268, 708–714. [Google Scholar] [CrossRef] - Thankachan, T.; Soorya Prakash, K.; Kamarthin, M. Optimizing the Tribological Behavior of Hybrid Copper Surface Composites Using Statistical and Machine Learning Techniques. J. Tribol. Trans. ASME
**2018**, 140, 031610. [Google Scholar] [CrossRef] - Sadık Ünlü, B.; Durmuş, H.; Meriç, C. Determination of tribological properties at CuSn10 alloy journal bearings by experimental and means of artificial neural networks method. Ind Lubr. Tribol.
**2012**, 64, 258–264. [Google Scholar] [CrossRef] - Senatore, A.; D’Agostino, V.; Di Giuda, R.; Petrone, V. Experimental investigation and neural network prediction of brakes and clutch material frictional behaviour considering the sliding acceleration influence. Tribol. Int.
**2011**, 44, 1199–1207. [Google Scholar] [CrossRef] - Bhaumik, S.; Mathew, B.R.; Datta, S. Computational intelligence-based design of lubricant with vegetable oil blend and various nano friction modifiers. Fuel
**2019**, 241, 733–743. [Google Scholar] [CrossRef] - Schwarz, S.; Grillenberger, H.; Graf-Goller, O.; Bartz, M.; Tremmel, S.; Wartzack, S. Using Machine Learning Methods for Predicting Cage Performance Criteria in an Angular Contact Ball Bearing. Lubricants
**2022**, 10, 25. [Google Scholar] [CrossRef] - Marian, M.; Mursak, J.; Bartz, M.; Profito, F.J.; Rosenkranz, A.; Wartzack, S. Predicting EHL film thickness parameters by machine learning approaches. Friction
**2022**, 11, 992–1013. [Google Scholar] [CrossRef] - Walker, J.; Questa, H.; Raman, A.; Ahmed, M.; Mohammadpour, M.; Bewsher, S.R.; Offner, G. Application of Tribological Artificial Neural Networks in Machine Elements. Tribol. Lett.
**2023**, 71, 3. [Google Scholar] [CrossRef] - Hess, N.; Shang, L. Development of a Machine Learning Model for Elastohydrodynamic Pressure Prediction in Journal Bearings. J. Tribol. Trans. ASME
**2022**, 144, 081603. [Google Scholar] [CrossRef] - Garabedian, N.T.; Schreiber, P.J.; Brandt, N.; Zschumme, P.; Blatter, I.L.; Dollmann, A.; Haug, C.; Kümmel, D.; Li, Y.; Meyer, F.; et al. Generating FAIR research data in experimental tribology. Sci. Data
**2022**, 9, 315. [Google Scholar] [CrossRef] - Brandt, N.; Garabedian, N.T.; Schoof, E.; Schreiber, P.J.; Zschumme, P.; Greiner, C.; Selzer, M. Managing FAIR Tribological Data Using Kadi4Mat. Data
**2022**, 7, 15. [Google Scholar] [CrossRef] - Bagov, I.; Greiner, C.; Garabedian, N. Collaborative Metadata Definition using Controlled Vocabularies, and Ontologies. RIO
**2022**, 8, e94931. [Google Scholar] [CrossRef] - Kügler, P.; Marian, M.; Dorsch, R.; Schleich, B.; Wartzack, S. A Semantic Annotation Pipeline towards the Generation of Knowledge Graphs in Tribology. Lubricants
**2022**, 10, 18. [Google Scholar] [CrossRef] - Kügler, P.; Marian, M.; Schleich, B.; Tremmel, S.; Wartzack, S. tribAIn—Towards an Explicit Specification of Shared Tribological Understanding. Appl. Sci.
**2020**, 10, 4421. [Google Scholar] [CrossRef] - Karniadakis, G.E.; Kevrekidis, I.G.; Lu, L.; Perdikaris, P.; Wang, S.; Yang, L. Physics-informed machine learning. Nat. Rev. Phys.
**2021**, 3, 422–440. [Google Scholar] [CrossRef] - Raissi, M.; Perdikaris, P.; Karniadakis, G.E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys.
**2019**, 378, 686–707. [Google Scholar] [CrossRef] - Raissi, M.; Karniadakis, G.E. Hidden physics models: Machine learning of nonlinear partial differential equations. J. Comput. Phys.
**2018**, 357, 125–141. [Google Scholar] [CrossRef] - Lagaris, I.E.; Likas, A.; Fotiadis, D.I. Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw.
**1998**, 9, 987–1000. [Google Scholar] [CrossRef] - Pioch, F.; Harmening, J.H.; Müller, A.M.; Peitzmann, F.-J.; Schramm, D.; el Moctar, O. Turbulence Modeling for Physics-Informed Neural Networks: Comparison of Different RANS Models for the Backward-Facing Step Flow. Fluids
**2023**, 8, 43. [Google Scholar] [CrossRef] - Almajid, M.M.; Abu-Al-Saud, M.O. Prediction of porous media fluid flow using physics informed neural networks. J. Pet. Sci. Eng.
**2022**, 208, 109205. [Google Scholar] [CrossRef] - Rudy, S.H.; Brunton, S.L.; Proctor, J.L.; Kutz, J.N. Data-driven discovery of partial differential equations. Sci. Adv.
**2017**, 3, e1602614. [Google Scholar] [CrossRef] - Chen, D.; Li, Y.; Liu, K.; Li, Y. A physics-informed neural network approach to fatigue life prediction using small quantity of samples. Int. J. Fatigue
**2023**, 166, 107270. [Google Scholar] [CrossRef] - Lee, S.; Popovics, J. Applications of physics-informed neural networks for property characterization of complex materials. RILEM Tech. Lett.
**2022**, 7, 178–188. [Google Scholar] [CrossRef] - Taç, V.; Linka, K.; Sahli-Costabal, F.; Kuhl, E.; Tepole, A.B. Benchmarking physics-informed frameworks for data-driven hyperelasticity. Comput. Mech.
**2023**. [Google Scholar] [CrossRef] - Pun, G.P.P.; Batra, R.; Ramprasad, R.; Mishin, Y. Physically informed artificial neural networks for atomistic modeling of materials. Nat. Commun.
**2019**, 10, 2339. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z.; Gu, G.X. Physics-informed deep learning for digital materials. Theor. Appl. Mech. Lett.
**2021**, 11, 100220. [Google Scholar] [CrossRef] - Katsikis, D.; Muradova, A.D.; Stavroulakis, G.E. A Gentle Introduction to Physics-Informed Neural Networks, with Applications in Static Rod and Beam Problems. J. Adv. App. Comput. Math.
**2022**, 9, 103–128. [Google Scholar] [CrossRef] - Moradi, S.; Duran, B.; Eftekhar Azam, S.; Mofid, M. Novel Physics-Informed Artificial Neural Network Architectures for System and Input Identification of Structural Dynamics PDEs. Buildings
**2023**, 13, 650. [Google Scholar] [CrossRef] - van Herten, R.L.M.; Chiribiri, A.; Breeuwer, M.; Veta, M.; Scannell, C.M. Physics-informed neural networks for myocardial perfusion MRI quantification. Med. Image Anal.
**2022**, 78, 102399. [Google Scholar] [CrossRef] - Sahli Costabal, F.; Yang, Y.; Perdikaris, P.; Hurtado, D.E.; Kuhl, E. Physics-Informed Neural Networks for Cardiac Activation Mapping. Front. Phys.
**2020**, 8, 42. [Google Scholar] [CrossRef] - Almqvist, A. Fundamentals of Physics-Informed Neural Networks Applied to Solve the Reynolds Boundary Value Problem. Lubricants
**2021**, 9, 82. [Google Scholar] [CrossRef] - Bach, F. Breaking the Curse of Dimensionality with Convex Neural Networks. J. Mach. Learn. Res.
**2014**, 18, 629–681. [Google Scholar] - Zhao, Y.; Guo, L.; Wong, P.P.L. Application of physics-informed neural network in the analysis of hydrodynamic lubrication. Friction
**2023**, 11, 1253–1264. [Google Scholar] [CrossRef] - Zubov, K.; McCarthy, Z.; Ma, Y.; Calisto, F.; Pagliarino, V.; Azeglio, S.; Bottero, L.; Luján, E.; Sulzer, V.; Bharambe, A.; et al. NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations. arXiv
**2021**, arXiv:2107.09443. [Google Scholar] - Li, L.; Li, Y.; Du, Q.; Liu, T.; Xie, Y. ReF-nets: Physics-informed neural network for Reynolds equation of gas bearing. Comput. Methods Appl. Mech. Eng.
**2022**, 391, 114524. [Google Scholar] [CrossRef] - Yadav, S.K.; Thakre, G. Solution of Lubrication Problems with Deep Neural Network. In Advances in Manufacturing Engineering; Dikshit, M.K., Soni, A., Davim, J.P., Eds.; Springer Nature Singapore: Singapore, 2023; pp. 471–477. ISBN 978-981-19-4207-5. [Google Scholar]
- Xi, Y.; Deng, J.; Li, Y. A solution for finite journal bearings by using physics-informed neural networks with both soft and hard constrains. Ind Lubr. Tribol.
**2023**, 75, 560–567. [Google Scholar] [CrossRef] - Rom, M. Physics-informed neural networks for the Reynolds equation with cavitation modeling. Tribol. Int.
**2023**, 179, 108141. [Google Scholar] [CrossRef] - Cheng, Y.; He, Q.; Huang, W.; Liu, Y.; Li, Y.; Li, D. HL-nets: Physics-informed neural networks for hydrodynamic lubrication with cavitation. Tribol. Int.
**2023**, 188, 108871. [Google Scholar] [CrossRef] - Swift, H.W. The Stability of Lubricating Films in Journal Bearings. Minutes Proc. Inst. Civ. Eng.
**1932**, 233, 267–288. [Google Scholar] - Stieber, W. Hydrodynamische Theorie des Gleitlagers. Das Schwimmlager; VDI: Berlin, Germany, 1933. [Google Scholar]
- Jakobsson, B.; Floberg, L. The Finite Journal Bearing, Considering Vaporization; Gumperts: Göteborg, Sweden, 1957. [Google Scholar]
- Olsson, K.-O. Cavitation in Dynamically Loaded Bearings; Gumperts: Göteborg, Sweden, 1965. [Google Scholar]
- Haviez, L.; Toscano, R.; El Youssef, M.; Fouvry, S.; Yantio, G.; Moreau, G. Semi-physical neural network model for fretting wear estimation. J. Intell. Fuzzy Syst.
**2015**, 28, 1745–1753. [Google Scholar] [CrossRef] - Yucesan, Y.A.; Viana, F.A.C. A Physics-informed Neural Network for Wind Turbine Main Bearing Fatigue. Int. J. Progn. Health Manag.
**2020**, 11. [Google Scholar] [CrossRef] - Shen, S.; Lu, H.; Sadoughi, M.; Hu, C.; Nemani, V.; Thelen, A.; Webster, K.; Darr, M.; Sidon, J.; Kenny, S. A physics-informed deep learning approach for bearing fault detection. Eng. Appl. Artif. Intell.
**2021**, 103, 104295. [Google Scholar] [CrossRef] - Ni, Q.; Ji, J.C.; Halkon, B.; Feng, K.; Nandi, A.K. Physics-Informed Residual Network (PIResNet) for rolling element bearing fault diagnostics. Mech. Syst. Signal Process.
**2023**, 200, 110544. [Google Scholar] [CrossRef] - Li, Y.; Wang, J.; Huang, Z.; Gao, R.X. Physics-informed meta learning for machining tool wear prediction. J. Manuf. Syst.
**2022**, 62, 17–27. [Google Scholar] [CrossRef] - Marian, M.; Almqvist, A.; Rosenkranz, A.; Fillon, M. Numerical micro-texture optimization for lubricated contacts—A critical discussion. Friction
**2022**, 10, 1772–1809. [Google Scholar] [CrossRef] - Shukla, K.; Jagtap, A.D.; Karniadakis, G.E. Parallel physics-informed neural networks via domain decomposition. J. Comput. Phys.
**2021**, 447, 110683. [Google Scholar] [CrossRef] - Dwivedi, V.; Srinivasan, B. Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations. Neurocomputing
**2020**, 391, 96–118. [Google Scholar] [CrossRef] - Wu, C.; Zhu, M.; Tan, Q.; Kartha, Y.; Lu, L. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Comput. Methods Appl. Mech. Eng.
**2023**, 403, 115671. [Google Scholar] [CrossRef]

**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 2.**Data and physics scenarios. Redrawn and adapted from [48].

**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 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 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 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**) L

_{2}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 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 9.**Semi-PINN two-level structure used to predict fretting wear. Reprinted from [31] with permission from CC BY 4.0.

**Figure 10.**Hybrid PINN for main bearing fatigue and grease degradation. Reprinted from [78] with permission from CC BY 3.0.

**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.

Field of Application | PIML Approach | Year | Reference |
---|---|---|---|

Lubrication prediction | Using PINN to solve the 1D Reynolds BVP to predict the pressure distribution in a fluid-lubricated linear converging slider | 2021 | [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 slider | 2023 | [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 bearing | 2022 | [68] | |

Using PINN to solve the 2D Reynolds equation to predict the behavior of fluid-lubricated journal as well as two-lobe bearings | 2023 | [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 viscosity | 2023 | [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 cavitation | 2023 | [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 cavitation | 2023 | [72] | |

Wear and damage prediction | Using semi PINN to find regression fitting parameters for Archard’s wear law based upon small data from fretting wear experiments | 2015 | [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 approach | 2020 | [78] | |

Using physics-informed CNN with preceding threshold model for rolling bearing fault detection | 2021 | [79] | |

Using physics-informed residual network for rolling bearing fault detection | 2023 | [80] | |

Using PIML framework consisting of piecewise fitting, a hybrid physics-informed data-driven model, and meta-learning to predict tool wear | 2022 | [81] |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Marian, M.; Tremmel, S.
Physics-Informed Machine Learning—An Emerging Trend in Tribology. *Lubricants* **2023**, *11*, 463.
https://doi.org/10.3390/lubricants11110463

**AMA Style**

Marian M, Tremmel S.
Physics-Informed Machine Learning—An Emerging Trend in Tribology. *Lubricants*. 2023; 11(11):463.
https://doi.org/10.3390/lubricants11110463

**Chicago/Turabian Style**

Marian, Max, and Stephan Tremmel.
2023. "Physics-Informed Machine Learning—An Emerging Trend in Tribology" *Lubricants* 11, no. 11: 463.
https://doi.org/10.3390/lubricants11110463