Reduced-order Surrogate Model Research Articles

Fast simulation of coronary in‐stent restenosis: A non‐intrusive data‐driven reduced order surrogate model

AbstractModeling and simulation of coronary artery disease (CAD) is of great importance for supporting and predicting the outcome of percutaneous coronary intervention (PCI). However, an in silico model generally requires heavy computational resources. An effective reduced order surrogate model is indispensable in this context. This study aims to develop a non‐intrusive data‐driven reduced order surrogate model for coronary in‐stent restenosis (ISR) incorporating anti‐inflammatory drugs embedded in the drug‐eluting stents. The constitutive model includes a detailed multiphysics approach based on partial differential equations (PDEs), which include descriptions of platelet aggregation, growth‐factor release, cellular motility and drug deposition. Dimensionality reduction is carried out based on a 3D convolutional autoencoder, which comprises an encoder and decoder. The former condenses the full‐order solution into a lower‐dimensional latent space, while the latter recovers the full solution from the latent space. Special attention is paid to handle the multidimensional outputs and network architecture.

Data-driven reduced order surrogate modeling for coronary in-stent restenosis

Background:The intricate process of coronary in-stent restenosis (ISR) involves the interplay between different mediators, including platelet-derived growth factor, transforming growth factor-β, extracellular matrix, smooth muscle cells, endothelial cells, and drug elution from the stent. Modeling such complex multiphysics phenomena demands extensive computational resources and time. Methods:This paper proposes a novel non-intrusive data-driven reduced order modeling approach for the underlying multiphysics time-dependent parametrized problem. In the offline phase, a 3D convolutional autoencoder, comprising an encoder and decoder, is trained to achieve dimensionality reduction. The encoder condenses the full-order solution into a lower-dimensional latent space, while the decoder facilitates the reconstruction of the full solution from the latent space. To deal with the 5D input datasets (3D geometry + time series + multiple output channels), two ingredients are explored. The first approach incorporates time as an additional parameter and applies 3D convolution on individual time steps, encoding a distinct latent variable for each parameter instance within each time step. The second approach reshapes the 3D geometry into a 2D plane along a less interactive axis and stacks all time steps in the third direction for each parameter instance. This rearrangement generates a larger and complete dataset for one parameter instance, resulting in a singular latent variable across the entire discrete time-series. In both approaches, the multiple outputs are considered automatically in the convolutions. Moreover, Gaussian process regression is applied to establish correlations between the latent variable and the input parameter. Results:The constitutive model reveals a significant acceleration in neointimal growth between 30−60 days post percutaneous coronary intervention (PCI). The surrogate models applying both approaches exhibit high accuracy in pointwise error, with the first approach showcasing smaller errors across the entire evaluation period for all outputs. The parameter study on drug dosage against ISR rates provides noteworthy insights of neointimal growth, where the nonlinear dependence of ISR rates on the peak drug flux exhibits intriguing periodic patterns. Applying the trained model, the rate of ISR is effectively evaluated, and the optimal parameter range for drug dosage is identified. Conclusion:The demonstrated non-intrusive reduced order surrogate model proves to be a powerful tool for predicting ISR outcomes. Moreover, the proposed method lays the foundation for real-time simulations and optimization of PCI parameters.

An adaptive learning strategy for surrogate modeling of high-dimensional functions - Application to unsteady hypersonic flows in chemical nonequilibrium

Many engineering applications rely on the evaluation of expensive, non-linear high-dimensional functions. In this paper, we propose the RONAALP algorithm (Reduced Order Nonlinear Approximation with Adaptive Learning Procedure) to incrementally learn, on-the-fly as the application progresses, a fast and accurate reduced-order surrogate model of a target function. First, a combination of nonlinear auto-encoder, community clustering, and radial basis function networks allows us to learn an efficient and compact surrogate model with limited training data. Secondly, an active learning procedure overcomes any extrapolation issues during the online stage by adapting the surrogate model with high-fidelity evaluations that fall outside its current validity range. This approach results in generalizable, fast, and accurate reduced-order models of high-dimensional functions. The method is demonstrated on three direct numerical simulations of hypersonic flows in chemical nonequilibrium. Accurate simulations of these flows rely on detailed thermochemical gas models that dramatically increase the cost of such calculations. Using RONAALP to learn a reduced-order thermodynamic model surrogate on-the-fly, the cost of such simulations was reduced by up to 75% while maintaining an error of less than 10% on relevant quantities of interest.

Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

Multi-fidelity reduced-order surrogate modelling

High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated. Multi-fidelity surrogate modelling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are scarce. However, low-fidelity models, while often displaying the qualitative solution behaviour, fail to accurately capture fine spatio-temporal and dynamic features of high-fidelity models. To address this shortcoming, we present a data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states—time-parameter-dependent expansion coefficients of the POD basis—using a multi-fidelity long short-term memory network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality of this method is demonstrated by a collection of PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features.

Active-learning-driven surrogate modeling for efficient simulation of parametric nonlinear systems

When repeated evaluations for varying parameter configurations of a high-fidelity physical model are required, surrogate modeling techniques based on model order reduction are desirable. In absence of the governing equations describing the dynamics, we need to construct the parametric reduced-order surrogate model in a non-intrusive fashion. In this setting, the usual residual-based error estimate for optimal parameter sampling associated with the reduced basis method is not directly available. Our work provides a non-intrusive error-estimator-based optimality criterion to efficiently populate the parameter snapshots, thereby, enabling us to effectively construct a parametric surrogate model. We consider parameter-specific proper orthogonal decomposition subspaces and propose an active-learning-driven surrogate model using kernel-based shallow neural networks (KSNNs), abbreviated as ActLearn-POD-KSNN surrogate model. The center location for each kernel, along with center-dependent kernel widths, can be learned for the KSNN by using an alternating dual-staged iterative training procedure. To demonstrate the efficiency of our proposed ideas, we present numerical experiments using four physical models, including incompressible Navier–Stokes equations. The ActLearn-POD-KSNN surrogate model efficiently predicts the solution at new parameter locations, even for a setting with multiple interacting shock profiles and a fluid flow scenario with Hopf bifurcation. We also provide an investigation of the surrogate’s performance when the available data is noisy.

A data-driven reduced-order surrogate model for entire elastoplastic simulations applied to representative volume elements

This contribution discusses surrogate models that emulate the solution field(s) in the entire simulation domain. The surrogate uses the most characteristic modes of the solution field(s), in combination with neural networks to emulate the coefficients of each mode. This type of surrogate is well known to rapidly emulate flow simulations, but rather new for simulations of elastoplastic solids. The surrogate avoids the iterative process of constructing and solving the linearized governing equations of rate-independent elastoplasticity, as necessary for direct numerical simulations or (hyper-)reduced-order-models. Instead, the new plastic variables are computed only once per increment, resulting in substantial time savings. The surrogate uses a recurrent neural network to treat the path dependency of rate-independent elastoplasticity within the neural network itself. Because only a few of these surrogates have been developed for elastoplastic simulations, their potential and limitations are not yet well studied. The aim of this contribution is to shed more light on their numerical capabilities in the context of elastoplasticity. Although more widely applicable, the investigation focuses on a representative volume element, because these surrogates have the ability to both emulate the macroscale stress-deformation relation (which drives the multiscale simulation), as well as to recover all microstructural quantities within each representative volume element.

Posterior Assessment of Parameters in a Time Domain Random Walk Model of Partitioning Tracer Tests in Two‐Phase Flow Scenarios

AbstractWe provide a reliable and efficient methodological framework for the interpretation of laboratory‐scale partitioning tracer test data under uncertainty. The proposed approach rests on a Time domain random walk (TDRW) particle tracking methodology. The range of applicability of the latter is extended to include transport of partitioning tracers upon considering retardation and trapping mechanisms. A classical maximum likelihood (ML) approach is applied considering the extensive set of experimental observations of Dwarakanath et al. (1999, https://doi.org/10.1021/es990082v). This yields best estimates of model parameters, including residual immobile phase saturation, the partition coefficient and the parameters of the memory function employed to simulate the impact of solute trapping. Experimental observations of the partition coefficient are included in the objective function upon relying on a regularization term. We show that considering these types of information, which are typically obtained through batch experiments, is important to attain joint estimates of the partition coefficient and of residual immobile phase saturation. Sample probability distributions of model parameters conditional on available data are then assessed through a stochastic inverse modeling approach. This step poses a significant challenge in terms of computational effort and is performed through a reduced order surrogate model. Our results show that the TDRW‐based approach can effectively capture the key features of the observed breakthrough curves of the various partitioning tracers analyzed and provide satisfactory estimates of residual immobile phase saturation.

Multi-Objective Optimization of Variable Density Multi-Layer Insulation for Liquid Hydrogen Containers Based on Reduced-Order Surrogate Model

For liquid hydrogen transportation, thermal insulation materials that are lightweight, compact and exhibit high-performance have been pursued for several decades, and variable density multi-layer insulation (VD-MLI) has been regarded as a promising choice. The thermal insulation performance of the insulation materials is important, but is not at the top of the list; many constraints, such as the space and weight of the insulation structures, are imposed on the design of a VD-MLI. Consequently, this makes the optimization of VD-MLIs more complicated. The present authors conducted a multi-objective optimization of a VD-MLI stacked with specific insulation units. The number of repetitions of the basic insulation unit was regarded as the dimensionless design parameter of the VD-MLI. Based on the experimentally validated layer-by-layer (LBL) model for MLI design, the multi-objective optimization of VD-MLI for liquid hydrogen storage was conducted by the combination of proper orthogonal decomposition with a general regression neural network (POD-GRNN) surrogate model optimization framework. The results showed that the optimal solutions for VD-MLI configurations could be achieved under different constraints. The present optimization framework provides a new reference for the optimization of VD-MLI for cryogenic liquid storage.

Towards the evaluation of heat and mass transfer in pipe flows with cocurrent falling films using One‐Dimensional Turbulence

AbstractTurbulent mixed convection in an air‐water system is evaluated with a novel numerical solver implementing the stochastic One‐Dimensional Turbulence (ODT) model in a turbulent air flow surrounded by a laminar cocurrent water falling film in a cylindrical geometry. The ODT model is used as a reduced order surrogate model for the effects of turbulent advection, turbulent heat flux, and turbulent mass‐flux within a one‐dimensional domain. An ad‐hoc temporal‐to‐spatial transformation relying on the bulk flow gas velocity is used to obtain streamwise‐dependent statistics of the flow. The ODT simulation results are compared to simulations obtained with the assumption of a quasi‐laminar one‐dimensional gas flow, and to Reynolds‐Averaged Navier‐Stokes (RANS) reference data for a cocurrent water falling film evaporator [1]. The results show that the turbulent transport plays a decisive role in the estimation of interface gradients of temperature and vapor mass fraction. Although ODT predicts global quantities such as the interface temperature in a reasonable way, the model falls short of successfully predicting streamwise‐dependent radial profiles. Despite the shortcomings, the framework presented here is the first stepping stone towards the evaluation of complex multiphase momentum, mass, and heat transfer couplings with full scale resolution on potential evaporative devices. The model, thus, provides valuable information with minimal empiricism on the dynamics of the small scales for pioneering engineering applications.

Radial basis function interpolation of fields resulting from nonlinear simulations

Three approaches for construction of a surrogate model of a result field consisting of multiple physical quantities are presented. The first approach uses direct interpolation of the result space on the input space. In the second and third approaches a Singular Value Decomposition is used to reduce the model size. In the reduced order surrogate models, the amplitudes corresponding to the different basis vectors are interpolated. A quality measure that takes into account different physical parts of the result field is defined. As the quality measure is very cheap to evaluate, it can be used to efficiently optimize hyperparameters of all surrogate models. Based on the quality measure, a criterion is proposed to choose the number of basis vectors for the reduced order models. The performance of the surrogate models resulting from the three different approaches is compared using the quality measure based on a validation set. It is found that the novel criterion can effectively be used to select the number of basis vectors. The choice of construction method significantly influences the quality of the surrogate model.

Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models

Reduced-order modelling and low-dimensional surrogate models generated using machine learning algorithms have been widely applied in high-dimensional dynamical systems to improve the algorithmic efficiency. In this paper, we develop a system which combines reduced-order surrogate models with a novel data assimilation (DA) technique used to incorporate real-time observations from different physical spaces. We make use of local smooth surrogate functions which link the space of encoded system variables and the one of current observations to perform variational DA with a low computational cost. The new system, named generalised latent assimilation can benefit both the efficiency provided by the reduced-order modelling and the accuracy of data assimilation. A theoretical analysis of the difference between surrogate and original assimilation cost function is also provided in this paper where an upper bound, depending on the size of the local training set, is given. The new approach is tested on a high-dimensional (CFD) application of a two-phase liquid flow with non-linear observation operators that current Latent Assimilation methods can not handle. Numerical results demonstrate that the proposed assimilation approach can significantly improve the reconstruction and prediction accuracy of the deep learning surrogate model which is nearly 1000 times faster than the CFD simulation.

Surrogate model for gravitational wave signals from nonspinning, comparable-to large-mass-ratio black hole binaries built on black hole perturbation theory waveforms calibrated to numerical relativity

We present a reduced-order surrogate model of gravitational waveforms from non-spinning binary black hole systems with comparable to large mass-ratio configurations. This surrogate model, \texttt{BHPTNRSur1dq1e4}, is trained on waveform data generated by point-particle black hole perturbation theory (ppBHPT) with mass ratios varying from 2.5 to 10,000. \texttt{BHPTNRSur1dq1e4} extends an earlier waveform model, \texttt{EMRISur1dq1e4}, by using an updated transition-to-plunge model, covering longer durations up to 30,500 $m_1$ (where $m_1$ is the mass of the primary black hole), includes several more spherical harmonic modes up to $\ell=10$, and calibrates subdominant modes to numerical relativity (NR) data. In the comparable mass-ratio regime, including mass ratios as low as $2.5$, the gravitational waveforms generated through ppBHPT agree surprisingly well with those from NR after this simple calibration step. We also compare our model to recent SXS and RIT NR simulations at mass ratios ranging from $15$ to $32$, and find the dominant quadrupolar modes agree to better than $\approx 10^{-3}$. We expect our model to be useful to study intermediate-mass-ratio binary systems in current and future gravitational-wave detectors.

Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics.

We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the "curse of dimensionality" related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to the original high-dimensional space using radial basis function interpolation and geometric harmonics. The proposed numerical data-driven scheme can also be applied as a reduced-order model procedure for the numerical solution/propagation of the (transient) dynamics of partial differential equations (PDEs). We assess the performance of the proposed scheme via three different families of problems: (a) the forecasting of synthetic time series generated by three simplistic linear and weakly nonlinear stochastic models resembling electroencephalography signals, (b) the prediction/propagation of the solution profiles of a linear parabolic PDE and the Brusselator model (a set of two nonlinear parabolic PDEs), and (c) the forecasting of a real-world data set containing daily time series of ten key foreign exchange rates spanning the time period 3 September 2001-29 October 2020.

Multifidelity Ensemble Kalman Filtering Using Surrogate Models Defined by Theory-Guided Autoencoders

Data assimilation is a Bayesian inference process that obtains an enhanced understanding of a physical system of interest by fusing information from an inexact physics-based model, and from noisy sparse observations of reality. The multifidelity ensemble Kalman filter (MFEnKF) recently developed by the authors combines a full-order physical model and a hierarchy of reduced order surrogate models in order to increase the computational efficiency of data assimilation. The standard MFEnKF uses linear couplings between models, and is statistically optimal in case of Gaussian probability densities. This work extends the MFEnKF into to make use of a broader class of surrogate model such as those based on machine learning methods such as autoencoders non-linear couplings in between the model hierarchies. We identify the right-invertibility property for autoencoders as being a key predictor of success in the forecasting power of autoencoder-based reduced order models. We propose a methodology that allows us to construct reduced order surrogate models that are more accurate than the ones obtained via conventional linear methods. Numerical experiments with the canonical Lorenz'96 model illustrate that nonlinear surrogates perform better than linear projection-based ones in the context of multifidelity ensemble Kalman filtering. We additionality show a large-scale proof-of-concept result with the quasi-geostrophic equations, showing the competitiveness of the method with a traditional reduced order model-based MFEnKF.

Surrogate convolutional neural network models for steady computational fluid dynamics simulations

A convolution neural network (CNN)-based approach for the construction of reduced order surrogate models for computational fluid dynamics (CFD) simulations is introduced; it is inspired by the approach of Guo, Li, and Iori [X. Guo, W. Li, and F. Iorio, Convolutional neural networks for steady flow approximation, in Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '16, New York, USA, 2016, ACM, pp. 481–490]. In particular, the neural networks are trained in order to predict images of the flow field in a channel with varying obstacle based on an image of the geometry of the channel. A classical CNN with bottleneck structure and a U-Net are compared while varying the input format, the number of decoder paths, as well as the loss function used to train the networks. This approach yields very low prediction errors, in particular, when using the U-Net architecture. Furthermore, the models are also able to generalize to unseen geometries of the same type. A transfer learning approach enables the model to be trained to a new type of geometries with very low training cost. Finally, based on this transfer learning approach, a sequential learning strategy is introduced, which significantly reduces the amount of necessary training data.

Inverse parameter estimation to predict material parameters of the\xa0Cowper\u2013Symonds constitutive equation in electrohydraulic forming process

The development of a reliable numerical simulation is essential for understanding high-speed forming processes such as electrohydraulic forming (EHF). This numerical model should be created based on the accurate material properties. However, dynamic material properties at strain rates exceeding 1000 s^{-1} cannot be easily obtained through an experimental approach. Thus, this study predicted two material parameters in the Cowper–Symonds constitutive equation based on inverse parameter estimation, such that the parameters predicted using the numerical simulation corresponded well with those obtained from the experimental results. The target material was a 1-mm-thick Al 5052-H32 sheet. The comparison target included the final deformation shape of the sheet in the EHF-free bulging test at three input voltages of 6, 7, and 8 kV. For the inverse parameter estimation, the posterior distribution for the two parameters included a likelihood and a prior distribution. For the likelihood construction, a reduced-order surrogate model was developed in advance to substitute the numerical simulation based on ordinary Kriging and principal component analysis. Moreover, the error distribution of the bulge height between the experiment and reduced-order surrogate model was obtained. The prior distribution at 7 kV was defined as a uniform distribution, and the posterior distribution at 7 kV was employed as a prior distribution at 6, 7, and 8 kV. Furthermore, Markov chain Monte Carlo sampling was employed and the Metropolis–Hastings algorithm was adopted to obtain the samples following the posterior distribution. After the autocorrelation calculation for sample independence, the lag with an autocorrelation of pm ,0.02 interval was selected and every lag^{mathrm {th}} sample was obtained. The total number of acquired samples was 10^{5}, and the mean values were calculated from the obtained samples. Consequently, the numerical simulation with mean values displayed good agreement with the experimental results.

Extended reduced-order surrogate models for scalar-tensor gravity in the strong field and applications to binary pulsars and gravitational waves

Statistically sound tests of scalar-tensor gravity theories in the strong-field regime usually involves computationally intensive calculations. In this study, we construct a reduced order surrogate model for the scalar-tensor gravity of Damour and Esposito-Far\`ese (DEF) with spontaneous scalarization phenomena developed for neutron stars (NSs). This model allows us to perform a rapid and comprehensive prediction of NS properties, including mass, radius, moment of inertia, effective scalar coupling, and two extra coupling parameters. We code the model in the pySTGROMX package, as an extension of our previous work, that speeds up the calculations at two and even three orders of magnitude and yet still keeps accuracy of $\sim1\%$. Using the model, we can calculate all the post-Keplerian parameters in the timing of binary pulsars conveniently, which provides a quick approach for us to place comprehensive constraints on the DEF theory. We perform Markov-chain Monte Carlo simulations with the model to constrain the parameters of the DEF theory with well-timed binary pulsars. Utilizing five NS-white dwarf and three NS-NS binaries, we obtain the most stringent constraints on the DEF theory up to now. Our work provides a public tool for quick evaluation of NSs' derived parameters to test gravity in the strong-field regime.

Various reduced-order surrogate models for fluid flow and mass transfer in human bronchial tree.

The bronchial tree plays a main role in the human respiratory system because the air distribution throughout the lungs and gas exchange with blood occur in the airways whose dimensions vary from several centimeters to micrometers. Organization of about 60,000 conducting airways and 33 million respiratory airways in a limited space results in a complex structure. Due to this inherent complexity and a high number of airways, using target-oriented dimensional reduction is inevitable. In addition, there is no general reduced-order model for various types of problems. This necessitates coming up with an appropriate model from a variety of different reduced-order models to solve the desired problem. Lumped formulation, trumpet, or typical path model of whole or parts of bronchial tree are frequently used reduced-order models. On the other hand, using any of these models results in underestimation of flow heterogeneity leading to inaccurate prediction of the systems whose mechanisms depend on the fluid heterogeneity. In this study, a simple robust model combining mechanistic and non-mechanistic modeling approaches of the bronchial tree is proposed which overcomes the limitations of the previous reduced-order models and gives the same results of a detailed mechanistic model for the first time. This model starts from an accurate multi-branching model of conducting and respiratory airways (i.e., the base model) and suggests a proxy model of conducting airway and reduced-order model of respiratory airways based on the base model to significantly reduce computational cost while retaining the accuracy. The combination of these models suggests various reduced-order surrogate models of the human bronchial tree for different problems. The applications and limitations of each reduced-order model are also discussed. The accuracy of the proposed model in the prediction of fluid heterogeneity has been examined by the simulation of multi-breath inert gas washout because the alveolar slope is the reflection of fluid heterogeneity where the computational time decreases from 121h (using the base model) to 4.8s (using the reduced-order model). A parallel strategy for solving the equations is also proposed which decreases run time by 0.18s making the model suitable for real-time applications. Furthermore, the ability of the model has been evaluated in the modeling of asthmatic lung as an instance of abnormal lungs, and in the modeling of O2-CO2 exchange as an instance of nonlinear reacting systems. The results indicate that the proposed model outperforms previous models based on accuracy, robustness, and run time.

Hybrid Approach for Health Monitoring of Mud Motor Fleet

A mud motor is a kind of positive displacement motor (PDM) that is used to transform the hydraulic energy of drilling fluids (mud) into mechanical energy. This mechanical energy enables the drill bit to cut the rock and drill a well. It is one of the key parts of downhole assembly that is placed in the drillstring to provide additional power to the bit while drilling as its power downhole output is still unmatched. Mud motor failure is a common and costly issue in drilling operations. A proper prediction of the failure as well as an estimation of the remaining useful life (RUL) are essential for timely downhole mud motor maintenance and drilling optimization.
 Until now, the oil and gas industry has lacked reliable procedures to monitor and maintain the health of mud motors, resulting in unnecessary maintenance costs as well as unpredictable and costly drilling failures. Recently, Schlumberger has addressed this problem with an industry-first prognostics and health management (PHM) solution, which not only estimates the health of the mud motor and tracks RUL, but also creates a new service for clients and provides a competitive advantage. Timely mud motor retirement and maintenance will ultimately reduce failures and NPT.
 The proposed PHM solution is suitable for real-time implementation and combines two different sterling algorithms for reliable prediction of possible problems with the mud motors. It enables the estimation of the mud motor health both on the system level with the entire mud motor (system level PHM model) and on the subcomponent level (power section PHM model) – the most critical component of the mud motor.
 The system-level algorithm model leverages both surface and downhole drilling data as well as mud motor characteristic curves to compute the severity of mud motor degradation. A special mud motor degradation indicator is defined. The indicator is calculated to evaluate the degree of power section decay at each time recorded from thousands of field jobs. The trends of the degradation with respect to drilling time and drilling distance are extracted for each motor job. Based on the study of large datasets, good correlation was observed between the mud motor degradation indicator and mud motor failures.
 The power section PHM model uses downhole measurements to estimate the RUL of the elastomer – the life-limiting component inside the power section. It is based on a high-fidelity model and uses a hybrid approach by combining a high-fidelity physics-based model of a power section and data-driven approaches with machine learning techniques. Machine learning methods were applied to derive a reduced order surrogate model (ROM) of power sections from the original physics-based models for real-time applications. This ROM outputs the estimation of performance and fatigue characteristics of the considered power section depending on the considered drilling conditions such as differential pressure, downhole temperature, flow rate, and mud compatibility. As the result, the model analyzes accumulative risk of fatigue failure and produces real-time health information for the power section as a percentage of the remaining lifespan.
 The new solution for mud motor PHM was successfully verified and tested in the field. Comparison of the predicted mud motor fatigue life with the actual observed postjob conditions and job failures demonstrated good results of the developed models. The PHM enables optimization of mud motor selection, drilling configuration, and maintenance operations by minimizing RUL uncertainties while facilitating rerun decisions and avoiding overmaintenance and premature retirements. The whole solution is currently being integrated into a drilling platform including the maintenance system, the well construction planning, and the execution. It maximizes the equipment usage with increased drilling performance without sacrificing reliability and enables optimal fleet management of a drilling process for revenue maximization.