Hybrid analysis and modeling for next generation of digital twins
HHybrid analysis and modeling for next generation ofdigital twins
Suraj Pawar , Shady E. Ahmed , Omer San , Adil Rasheed School of Mechanical & Aerospace Engineering, Oklahoma State University, Stillwater, OK74078, USA. Department of Engineering Cybernetics, Norwegian University of Science and Technology,7465 Trondheim, Norway.E-mail: [email protected], [email protected], [email protected],[email protected]
Abstract.
The physics-based modeling has been the workhorse for many decades in manyscientific and engineering applications ranging from wind power, weather forecasting, andaircraft design. Recently, data-driven models are increasingly becoming popular in manybranches of science and engineering due to their non-intrusive nature and online learningcapability. Despite the robust performance of data-driven models, they are faced with challengesof poor generalizability and difficulty in interpretation. These challenges have encouraged theintegration of physics-based models with data-driven models, herein denoted hybrid analysisand modeling (HAM). We propose two different frameworks under the HAM paradigm forapplications relevant to wind energy in order to bring the physical realism within emergingdigital twin technologies. The physics-guided machine learning (PGML) framework reduces theuncertainty of neural network predictions by embedding physics-based features from a simplifiedmodel at intermediate layers and its performance is demonstrated for the aerodynamic forceprediction task. Our results show that the proposed PGML framework achieves approximately75% reduction in uncertainty for smaller angle of attacks. The interface learning (IL)framework illustrates how different solvers can be coupled to produce a multi-fidelity modeland is successfully applied for the Boussinesq equations that govern a broad class of transportprocesses. The IL approach paves the way for seamless integration of multi-scale, multi-physicsand multi-fidelity models ( M models).
1. Introduction
Some of the major challenges in realizing the potential of wind energy to meet the globalelectricity demand are the need for a deeper understanding of the physics of the atmospheric flow,science, and engineering of these large dynamic rotating machines and synergistic optimizationand control of fleets of wind farms within the electricity grid [1]. The decades of researchand development in fluid dynamics, systems engineering, manufacturing processes, materialdiscovery can now be complemented with unprecedented amounts of data generated from in-situ measurements, lab experiments, and numerical simulations to tackle these challenges. Thecombination of physics-based and data-driven models is increasing in every branch of scienceleading to the hybrid analysis and modeling (HAM) approach for many scientific applications.The HAM paradigm can be applied to a variety of tasks related to wind research, such asdigital twinning for the optimization and real-time control of wind farms. A digital twin is a r X i v : . [ phy s i c s . c o m p - ph ] J a n efined as a virtual replica of a physical system enabled through data collected from sensorsand simulations in real-time to solve problems such as control, optimization, monitoring,and improved decision-making [2]. The advancement in multiphysics solver, computationalinfrastructure, big data, and artificial intelligence has allowed us to make a digital replica of largephysical systems such as wind farms [3]. The digital twin lets us examine what if scenarios,evaluate the system’s response, and select the corresponding mitigation strategies. Figure 1shows the typical digital twin framework for a wind farm. The digital twin of wind farms can beuseful for different purposes such as optimal control of wind turbines to achieve the maximumperformance, routine maintenance of the equipment, accurate forecast of the power production,wind farm optimization [4], and improved decision-making. Figure 1.
Digital twin of the wind farm can allow us to evaluate different scenarios usinghybrid models. The big data collected from IoT sensors can be continuously assimilated tocorrect hybrid models and improve the state estimation.The success of the digital twins depends upon the type of approach that we employ formodeling the system. The ability of the HAM to combine the generalizability of physics-basedapproaches and the automatic pattern-identification feature of data-driven approaches makes itan attractive choice to model virtual replica of physical systems in digital twins. For example,there are several turbine wake models that capture the flow in the wake of a single turbine andthese models predict sufficiently accurate aerodynamics of a single turbine [5, 6]. However, thesemodels are not enough to capture the flows in wind farm due to wake superposition, complexterrains, deep array effects, and neglected physics [7, 8, 9]. The observational data collectedfrom a variety of sensors can be assumed to comprise of these complex flow interactions and themanifestation of all physical processes. Therefore, the hybrid model will aid in the robustnessof the digital twin rather than a pure physics-based or pure data-driven approach.In this work, we introduce two different frameworks under the umbrella of HAM. Thefirst framework is called physics-guided machine learning (PGML) where information fromsimplified physics-based models is incorporated within neural network architectures to improvethe generalizability of data-driven models [10]. The second framework is our interface learningapproach that deals with coupling different solvers and mathematical descriptions using2tatistical inference tools [11, 12, 13]. One of the potential applications of this framework canbe to specify the physically accurate inlet boundary condition for the wind farm simulation. Forexample, it is very common to use a coarse-grid solver to resolve the atmospheric boundary layerflow and employ the fine-grid solver to resolve the flow field around wind turbines in the windfarm. For such problems, it is very important to exchange the information between two solversthat leads to physically consistent boundary conditions, and statistical tools like deep learningcan be exploited for this.Concisely, bringing physical realism in digital twins will need • new modelling approaches that are accurate and certain, generalizable, computationallyefficient, and trustworthy, • seamless integration of multi-scale, multi-physics and multi-fidelity models ( M models).To this end, we propose two HAM approaches that address the above needs. While the firstapproach PGML provides a mechanism to guide the learning of a machine learning algorithmusing simplified physics, the second approach IL enables coupling of M models.
2. Physics-guided machine learning
A wide variety of problems in wind energy such as aerodynamic performance prediction [14, 15]for optimal control, design optimization, uncertainty quantification requires the prediction of thequantity of interest in real-time. The prediction of flow around an airfoil is a high-dimensional,and nonlinear problem that can be solved using high-fidelity methods like computationalfluid dynamics (CFD). However, these methods are computationally infeasible for real-timeprediction. On one hand, in certain flow regimes, the simplified methods like panel codescome with a non-negligible difference between the actual dynamics and approximate models forreal-world problems. On the other hand, the full-order CFD simulations are computationallydemanding, thus limiting their use in many inverse modeling methodologies that require amodel run to be performed in each iteration. To overcome these challenges, combining CFDmodels with machine learning to build a non-intrusive surrogate model is gaining widespreadpopularity [16, 17]. One of the main challenges with these non-intrusive models is theirpredictive capabilities for unseen data and their interpretation. Even though there are methodsto predict the uncertainty estimate in the prediction of machine learning models [18, 19], thegeneralizability of non-intrusive models is not on par with physics-based models. Meanwhile,the simplified models like the Blasius boundary layer model in fluid mechanics are highlygeneralizable across different conditions. Therefore, it is important to incorporate the domainknowledge into learning, and to this end, we exploit the relevant physics-based features frompanel methods through the PGML framework to enhance the generalizability of data-drivensurrogate model.We now introduce different components of the PGML framework depicted in Figure 2(a). Insupervised machine learning, the input vector x ∈ R m is fed to the machine learning model (forexample, the neural network in our case), and the mapping from the input vector to the outputvector y ∈ R n is learned through training. The neural network is trained to learn the function F θ , parameterized by θ , that includes the weights and biases of each neuron. The parametersof the neural network are optimized using the backpropagation algorithm to minimize the costfunction. Usually, for the regression problems, the cost function is the mean squared errorbetween true and predicted output, i.e., C ( x , θ ) = || y − F θ ( x ) || . In the PGML framework,the neural network is augmented with the output of the simplified physics-based model. Thefeatures extracted from simplified physics-based models are embedded into hidden layers alongwith latent variable. This is in contrast to admitting physics-based features at the input layerin conventional neural network architectures, which might lead to underestimation of the effectof such physical information, especially for high dimensional systems. For example, imagine3 achine LearningModel OutputInputSimplified Theory A B
PhysicsEnhancement CorrectorPredictor H i dden l a y e r H i dden l a y e r H i dden l a y e r H i dden l a y e r Physical parameters of the flow Prediction from the Hess-Smith panel method (a) (b)
Figure 2.
Physics guided machine learning (PGML) framework to train a learning enginebetween processes A and B : (a) a conceptual PGML framework which shows a predictor-corrector approach to incorporate physics into machine learning models through embeddingsimplified theories directly into neural network models, and (b) the representative neural networkarchitecture of the PGML framework used in this study for aerodynamic forces prediction task.stacking a few parameters to an input vector of a dimension of O (10 − ). It is highly probablethat the learning algorithm overlooks the effect of such prior knowledge in the minimizationalgorithm, and a modification of the cost function becomes necessary. In the PGML, a properlatent space is first identified and the information from the physics-based model aids the neuralnetwork in constraining its output to a manifold of physically realizable models.The training data for the neural network is generated using a series of numerical simulationsperformed in XFOIL [20]. We highlight here that the the neural network can also be trainedusing the data gathered from CFD simulations or wind-tunnel experiments. The lift coefficientdata were obtained for different Reynolds numbers Re between 1 × and 4 × and severalangles of attacks α in the range of −
20 to +20. A total of 168 sets of two-dimensional airfoilgeometry were generated from NACA 4-digit, NACA210, NACA220, and NACA250 series fortraining the neural network. Each airfoil is represented by 201 points. The maximum thicknessof all airfoils in the training dataset was between 6% to 18% of the chord length. We use theNACA23012 and NACA23024 airfoil geometry as the test dataset to evaluate the predictivecapability of the trained neural network. The simplified model used to generate the physics-based feature corresponds to the Hess-Smith panel method [21] based on potential flow theory.We note here that our testing airfoils are selected not only from a different NACA230 series(i.e., not used in the training dataset), but also the maximum thickness of 24% is well beyondthe thickness ratio limit included in the training dataset.The adopted neural network architecture has four hidden layers with 20 neurons in eachhidden layer. The physical parameters, i.e., the Reynolds number and the angle of attack areconcatenated at the third hidden layer along with the latent variables at that layer. In thePGML model, we augment the latent variables at the third layer with the lift coefficient and thepressure drag coefficient predicted by the panel method along with physical parameters of theflow (i.e., the Reynolds number and angle of attack). Therefore, the third layer of the neuralnetwork in the PGML framework has 24 latent variables. The representative neural network forthe PGML framework to predict the aerodynamic forces on an airfoil is displayed in Figure 2(b).We utilize an ensemble of neural networks trained using different initialization to predict theepistemic uncertainty [22, 23]. The weights and biases of each model are initialized using theGlorot uniform initializer and different random seed numbers are used to ensure that differentvalues of weights and biases are assigned for each model. The ensemble of all these models4ndicates the model uncertainty estimate of the predicted lift coefficient. Figure 3 shows theactual and predicted lift coefficient for the NACA23012 and NACA23024 airfoil geometry. Thereference
True performance is obtained by XFOIL. The ML corresponds to a simple feed-forwardneural network that uses the airfoil x and y coordinates as the input features, and the physicalparameters of the flow are concatenated at the third hidden layer along with the latent variablesat that layer. − −
10 0 10 20 α − . − . − . − . . . . . . C L NACA23012
TrueML − −
10 0 10 20 α − . − . − . − . . . . . . NACA23024
TrueML − −
10 0 10 20 α − . − . − . − . . . . . . C L NACA23012
TruePGML − −
10 0 10 20 α − . − . − . − . . . . . . NACA23024
TruePGML
Figure 3.
Actual versus predicted lift coefficient ( C L ) for NACA23012 and NACA23024 airfoilsat Re = 3 × using ML and PGML framework. The dashed blue curve represent theaverage of the predicted lift coefficient by all data-driven models (i.e., testing runs with differentinitialization seeds).As shown in Figure 3, we can see that the uncertainty in the prediction of the lift coefficient bythe PGML model is considerably less than the ML model for both NACA23012 and NACA23024airfoils. The proposed PGML framework provides significantly more accurate predictions withuncertainty reduced approximately by 75% especially for the angle of attacks between -10 and+12 degrees. This further illustrates the viability of the proposed PGML framework, sincethe physics embedding considered here employs constant source panels and a single vortex toapproximate the potential flow around the airfoil. We can also notice that the uncertaintyis higher for the angle of attacks outside the range of -10 to +12 degrees. This finding isnot surprising as the Hess-Smith panel method is a proven method for analysis of inviscidflow over airfoil for the smaller angle of attacks regime. The maximum thickness of an airfoilincluded in the training dataset is 18% of the chord length. Therefore, the uncertainty in theprediction of the lift coefficient by the ML model is higher for the NACA23024 airfoil comparedto the NACA23012 airfoil. These results clearly show the potential of the PGML framework forbuilding trustworthy models that can enable the digital twin of physical systems.5 . Interface learning The second framework we are introducing under the umbrella of HAM is the interface learning(IL). Multi-scale, multi-physics and multi-fidelity models ( M models) are the main beneficiariesfrom the IL methodology. Many complex systems relevant to scientific and engineeringapplications include multiple spatiotemporal scales and comprise a multifidelity problem sharingan interface between various formulations or heterogeneous computational entities. We refer thereaders to our previous discussion about the potential of IL approaches, with demonstrationson truncated domains [12], and mico-macro scale solvers coupling [11]. In the present study,we are interested in situations where part of the domain, physics, or scales are characterized byrepeating coherent structures, and can thus be represented by a reduced order model (ROM) forcomputational speed-up. In the meantime, a high-fidelity full order model (FOM) is dedicatedfor the rest of domain/dynamics for accuracy requirements. However, both solvers are coupledand information should be communicated and matched at their interface . To this end, we presenta robust HAM approach combining a physics-based FOM and a data-driven ROM to form thebuilding blocks of an integrated approach among mixed fidelity descriptions toward predictivedigital twin technologies, as depicted in Figure 4(a). Figure 4.
Hybrid analysis and modeling (HAM) as a key enabler for ROM-FOM couplingproblems toward predictive digital twins: (a) an overview, and (b) proposed ROM-FOM couplingframeworks.
In order to demonstrate the ROM-FOM coupling framework, we consider a coupled system asfollows, ∂u∂t = f ( u ; µ ) + g ( u, v ; µ , µ ) , (1) ∂v∂t = f ( v ; µ ) + g ( u, v ; µ , µ ) , (2)where u and v are the coupled variables and g and g define this coupling, while µ and µ denote the set of system’s parameters. We highlight that the coupled variables might representthe state variables at different regions of the domain (e.g., multi-component systems), differentphysics (e.g., fluid-structure interactions) and/or different scales within the same domain (e.g.,multiscale systems). We suppose that the dynamics of u can be approximated by a ROM whilea FOM resolves v and both solvers need to communicate information to satisfy the coupling.6 .1.1. Reduced order model Introducing a spatial discretization to Eq. (1), it can be rewrittenin a semi-discrete continuous-time as follows,d u d t = F ( u , v ; µ ) = L u + L v + N ( u , v ) , (3)where the boldfaced symbols represent the arrangement of discretized variables in a columnvector, µ defines the system’s parameters, and F is a deterministic operator with linear andnonlinear components L , and N , respectively. We exploit the advances and developments ofROM techniques to build surrogate models to economically resolve portions of domain and/orphysics. The ROM solution can thus be used to infer the flow conditions at the interfaceso that a FOM solver can be efficiently employed for the regions of interest. The standardGalerkin ansatz is applied for the dynamics of u as u ( t ) ≈ Φ α ( t ), where the columns of matrixΦ = [ φ , φ , . . . , φ r ] form the orthonormal bases of a reduced subspace, and α defines theiramplitudes. Proper orthogonal decomposition (POD) is one popular technique to systematicallyconstruct Φ such that the solution manifold preserves as much variance as possible whenprojected onto the subspace spanned by Φ [24]. By substituting this approximation into Eq. (3),performing the inner product with Φ, and making use of the quadratic nonlinearity in most fluidflow systems, we get the following, d α d t = L α + α T N α + C , (4)where L and N signify the model coefficients while C defines the contribution of v into theROM of u . At the interface, we introduce a long short-term memory network to bridge thelow-fidelity descriptions to high-fidelity models in various forms of interfacial error correctionor prolongation. An array of interface modeling paradigms are sketched in Figure 4(b) andsummarized as follows (see [13] for more details),(i) DPI: Direct Prolongation Interface.
The DPI approach provides an estimate of the flowvariables at the interface from the ROM solution. Indeed, this prolongation map naturallyresults from the Galerkin ansatz, without any interference from the ML side. However, it isknown that truncated Galerkin ROMs might yield erroneous and even unstable predictionsfor complex systems [25]. Therefore, the solution from the DPI approach is potentiallyinaccurate, and a correction needs to be introduced.(ii)
CPI: Correction followed by Prolongation Interface.
The CPI methodology works byintroducing the correction in the latent subspace and addresses the deviation in modalcoefficients predicted from solving the Galerkin ROM, known as closure error. Specifically,the LSTM for CPI takes the values of modal coefficients acquired from integrating Eq. (4)and predicts the discrepancy between these values and their optimal values. We highlighthere that the size of the input and output vectors is O ( r ), independent of the FOMresolution, which offers a potential flexibility dealing with 2D and 3D problems.(iii) UPI: Uplifted Prolongation Interface.
Although the CPI methodology cures the closureerror and provides a stabilized solution, it does not address the projection error. Unless alarge number of modes are resolved, the projection error can be significant. An upliftingROM has been proposed [25], where both closure and projection errors are taken care of.In addition to the closure modeling, the ROM subspace is expanded to recover some of thesmaller scales missing in the initial subspace as follows, u ≈ Φ α + Ψ β , (5)where Ψ forms orthonormal basis for a q -dimensional subspace complementing that spannedby Φ and with β being the corresponding amplitudes. We highlight that the Galerkin ROM7quations only solve for α to keep the computational cost as low as possible. Therefore,a complementary model for β has to be constructed so that the uplifting approach can beemployed. A mapping from the first r modal coefficients to the next q modes is assumedto exist and we exploit the LSTM learning capabilities to infer this map from data. Inparticular, the UPI architecture is trained to read the Galerkin ROM prediction for thefirst r modal coefficients as input, and return the true coefficients of the first r + q modes.Thus, it provides a closure correction for the first r modes and a superresolution effect forthe next q modes, simultaneously in a single network. The dimensionless form of the 2D incompressible Boussinesq equations can be represented bythe following two coupled transport equations in vorticity-streamfunction formulation, ∂ω∂t + J ( ω, ψ ) = 1Re ∇ ω + Ri ∂θ∂x , (6) ∂θ∂t + J ( θ, ψ ) = 1RePr ∇ θ, (7)where ω , ψ and θ denote the vorticity, streamfunction and temperature fields, respectively.We utilize the 2D Boussinesq equation to illustrate the ROM-FOM coupling in multi-fidelityenvironments. In particular, we suppose that we are more interested in the temperature fieldpredictions. Thus, we dedicate a FOM solver for Eq. (7). However, the solution of this equationrequires evaluating the streamfunction field at each time step. The kinematic relationshipbetween vorticity and streamfunction is given by the Poisson equation (i.e, ∇ ψ = − ω ), thesolution of which consumes significant amount of time and computational resources and isconsidered the bottleneck for most incompressible flow solvers. Therefore, we consider a ROMsolver for the voriticity dynamics and FOM solver for the temperature field.For demonstration, we explore the lock-exchange problem, defined by two fluids of differenttemperatures, in a rectangular domain ( x, y ) ∈ [0 , × [0 ,
1] with a vertical barrier dividing thedomain at x = 4, keeping the temperature of the left half at 1 . t = 0. Reynolds number of Re = 10 , Richardson number of Ri = 4,and Prandtl number of Pr = 1 are set and a Cartesian grid of 4096 ×
512 with a timestep of∆ t = 5 × − are used for the FOM simulations.A two-layer LSTM with 20 hidden units in each LSTM cell constitutes our ML architecture.We store 800 time snapshots for POD basis construction and we retain r = 8 modes for theGalerkin ROM solver. We also utilize the dataset of the stored 800 snapshots for LSTM trainingand validation. During the testing phase, the trained neural networks are deployed at each andevery timestep. This corresponds to the deployment of the presented approached 16000 times.Figure 5 shows the predictions of the temperature field at final time (i.e., t = 8) computed fromDPI, CPI, and UPI approaches compared to the FOM field. We emphasize that the ROM-FOMcoupling results correspond to the solution of the vorticity equation with a ROM solver, whichfeeds the FOM solver with streamfunction to solve the temperature equation only as opposed tothe FOM results which comes from the solution of both equations using a fully FOM simulation.Although the CPI results are better than those of DPI, we can observe that the fine details ofthe flow field are not accurately captured. On the other hand, the implementation of the UPIapproach with r = 8 and q = 8 recovers an increased amount of the fine flow structures that arenot well-represented by the first r = 8 modes. 8 igure 5. Final temperature fields as obtained from different ROM-FOM coupling approaches,compared to the FOM solution.
4. Concluding remarks
The data-driven methods are increasingly being applied in many branches of science andengineering due to their success in automatic pattern-identification using the data collectedfrom sensors and numerical simulations. Even though they offer an alternative to physics-basedmodeling derived from phenomenological arguments, they are usually black box in nature andlack generalizability. The hybrid analysis and modeling (HAM) is a newly emerging paradigmthat combines physics-based and data-driven modeling to deliver robust and generalizable modelsthat can enable the digital twins of large scale physical systems.The major contributions of this work towards HAM paradigm are • A novel deep neural network architecture that makes it possible to inject physics duringthe training process. This resulted in a significant reduction of uncertainty. • An interface learning technique that makes seamless coupling of multi fidelity modelspossible.The physics-guided machine learning (PGML) framework enhances the generalizability ofthe neural network based surrogate model and its performance is illustrated for real timeaerodynamic performance prediction task. The interface learning (IL) framework allowsintegration of two different models seamlessly to build multi-fidelity models that are ordersof magnitude faster than full order physics-based models. Although PGML approach wasdemonstrated to work with deep neural network, it can easily be extended to other machinelearning algorithms. Likewise the interface learning technique that was demonstrated to couplemulti-fidelity models can be used for coupling multiscale and multiphysics models.
Acknowledgement
The authors acknowledge the financial support from the Department of Engineering NorwegianResearch Council, the industrial partners of OPWIND: Operational Control for Wind PowerPlants (Grant No.: 268044/E20), the U.S. DOE Early Career Research Program (Award NumberDE-SC0019290), and the National Science Foundation under Award Number DMS-2012255.9 eferences [1] Paul Veers, Katherine Dykes, Eric Lantz, Stephan Barth, Carlo L Bottasso, Ola Carlson, Andrew Clifton,Johney Green, Peter Green, Hannele Holttinen, et al. Grand challenges in the science of wind energy.
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