A nudged hybrid analysis and modeling approach for realtime wake-vortex transport and decay prediction
Shady Ahmed, Suraj Pawar, Omer San, Adil Rasheed, Mandar Tabib
AA NUDGED HYBRID ANALYSIS AND MODELING APPROACH FORREALTIME WAKE - VORTEX TRANSPORT AND DECAY PREDICTION
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Shady E. Ahmed
School of Mechanical & Aerospace Engineering,Oklahoma State University,Stillwater, OK 74078, USA. [email protected]
Suraj Pawar
School of Mechanical & Aerospace Engineering,Oklahoma State University,Stillwater, OK 74078, USA. [email protected]
Omer San
School of Mechanical & Aerospace Engineering,Oklahoma State University,Stillwater, OK 74078, USA. [email protected]
Adil Rasheed
Department of Engineering Cybernetics,Norwegian University of Science and Technology,N-7465, Trondheim, Norway. [email protected]
Mandar Tabib
CSE Group, Mathematics and Cybernetics,SINTEF Digital,7034, Trondheim, Norway [email protected] A BSTRACT
In this paper, we put forth a long short-term memory (LSTM) nudging framework for the enhance-ment of reduced order models (ROMs) of fluid flows utilizing noisy measurements for air trafficimprovements. Toward emerging applications of digital twins in aviation, the proposed approachallows for constructing a realtime predictive tool for wake-vortex transport and decay systems. Webuild on the fact that in realistic application, there are uncertainties in initial and boundary conditions,model parameters, as well as measurements. Moreover, conventional nonlinear ROMs based onGalerkin projection (GROMs) suffer from imperfection and solution instabilities, especially foradvection-dominated flows with slow decay in the Kolmogorov width. In the presented LSTM nudg-ing (LSTM-N) approach, we fuse forecasts from a combination of imperfect GROM and uncertainstate estimates, with sparse Eulerian sensor measurements to provide more reliable predictions ina dynamical data assimilation framework. We illustrate our concept by solving a two-dimensionalvorticity transport equation. We investigate the effects of measurements noise and state estimateuncertainty on the performance of the LSTM-N behavior. We also demonstrate that it can sufficientlyhandle different levels of temporal and spatial measurement sparsity, and offer a huge potential indeveloping next-generation digital twin technologies. K eywords Nudging, Wake vortex, Separation distance, LSTM, Data assimilation, Reduced order modeling, Galerkinprojection, Measurement noise.
Aircraft wings are optimized to produce maximum lift and minimum drag. The design ensures that there is a high-pressure zone below the wing and a low-pressure zone above. Owing to this pressure gradient, air from below the wingis drawn around the wingtip into the region above the wing causing a vortex to trail from each wing tip. These wakevortices (WVs) are stable under calm atmospheric conditions and remain present in the free atmosphere for a very long a r X i v : . [ phy s i c s . c o m p - ph ] A ug PREPRINT time, retaining its shape and energy [1–3]. Furthermore, at very low altitudes, they might rebound from the ground andlinger on in the flight path corridor, posing significant risks to other aircraft that might encounter them.Small general aviation aircraft create vortices that are almost undetectable by a trailing aircraft of a similar size. Largejetliners, however, leave vortices that can exceed 150 mph in rotational velocity and are still detectable over distancesof 20 miles. The strongest and the most dangerous vortices are generated by aircraft that are heavy, in a clean gearand flaps-up condition, and flying at low speeds like those of landing approaches. WVs can cause violent rollingmotions and even flip a small aircraft upside down when a pilot trailing a large aircraft flies into the vortices [4]. Thepower of these dangerous spinning vortices can cause an aircraft to become uncontrollable. In almost all cases, controlof the aircraft can be restored if there is sufficient altitude. Some aircraft have sustained serious structural damagewhen encountering WVs, but were landed safely. There are, however, cases in which smaller trailing aircraft, during aclimb-out after take-off or during a landing approach, have crashed after entering WVs because they were too close tothe ground for the pilots to recover full control.It is due to the hazards posed by WVs left behind by a taking off or landing aircraft that serious precautions are to betaken. The operational minimum aircraft separation for different weight class configurations, used by the Air TrafficControl (ATC), varies from 2.5 to 6 nautical miles. However, when deciding the separation distance following thoseguidelines, the weather conditions and associated transport and decay of WVs are not taken into account. This wasnot a serious issue a couple of decades ago, but with the significant increase of the air traffic and a push for remotetowers for cost effective and safe operation, major airports around the world are feeling the pressure. In this regard,Digital Twins of airports appear like a potential solution; however, to realize such a digital twin will need accuratedescription of the prevailing condition at the airport as well as its evolution in the near foreseeable future. In the currentcontext, there is a need to develop a more efficient wake turbulence separation consisting of time-based minima betweendifferent aircraft types which takes into account the dynamic meteorological factors along with the variation in the wakegeneration mechanism associated with different class of aircraft. Such information will enable air traffic controllers todeliver consistent and safe spacing between aircraft leading to increased airport capacity, enhanced safety, reduced fuelconsumption, improved predictability and increased resilience.While the current solutions range from actively modifying/dissipating the wake-vortices using physical devices [5, 6]to accurately estimating the strengths of the vortices using LIDARS and RADARS [2, 7]. One shortfall of the twoapproaches is that none of them predicts the evolution of the vortices in the future. This gaps is being filled byadvanced computational fluid dynamics modeling which involves solving the highly non-linear Navier Stokes equationsat varying levels of approximations. However, their utility owing to their computationally demanding nature hasbeen limited to offline simulations geared towards developing a better understanding of the WV dynamics. At themoment, most of the fast WV models that are state-of-the art in WV predictive systems use physics-based empiricalparameterizations to mimic vortex transport and decay. Unfortunately, the computational efficiency of the fast WVmodels comes at the expense of accuracy. A good overview of the models can be found in [8]. To alleviate theproblems associated with the existing WV models, data-driven machine learning methods might appear attractive ata first glance, but their limited interpretability owing to their black-box nature make them a misfit for the kind ofsafety-critical application under consideration. To this end, building upon our recent works on the hybrid analysis andmodeling (HAM) framework [9–11], we present a data assimilation-empowered approach to utilize a machine learningmethodology to fuse computationally-light physics-based models with the available real-time measurement data toprovide more accurate and reliable predictions of wake-vortex transport and decay. In particular, we build a surrogatereduced order model (ROM), by combining proper orthogonal decomposition (POD) for basis construction [12–17] andGalerkin projection to model the dynamical evolution on the corresponding low-order subspace [18–26]. AlthoughROMs based on Galerkin projection (denoted as GROMs in the present study) have been traditionally considered thestandard approach for reduced order modeling, they often become inaccurate and unstable for long-term predictions ofconvection-dominated flows with strong nonlinearity [27–31]. Ideas like closure modeling [32–53] and Petrov-Galerkinprojection [54–62] have been investigated to address this deficiency. Alternatively, we exploit the nudging method [63]as a data assimilation (DA) framework, which works by relaxing the model state toward observations by addingcorrection (or nudging) terms, proportional to the difference between observations and model state, known as innovationin DA context. In classical DA nudging, this proportionality is assumed to be linear, and the proportionality constants (orweights) are empirically tuned. Instead, we introduce the hybridization at this stage, using a simplistic long short-termmemory (LSTM) architecture to generalize this relation to consider nonlinear mappings among the innovation andnudging terms.In other words, we utilize LSTM to combine the possibly defective model prediction with noisy measurements to“nudge” the model’s solution towards the true states [64, 65]. We apply the proposed LSTM nudging framework(denoted LSTM-N) for the reduced order modeling of the two-dimensional wake vortex problem in order to accuratelypredict the transport and decay of wake vortices for ATC applications. Moreover, we suppose that both inputs (i.e., thephysics-based model and data) are imperfect, thus avoiding biases in predictions. GROMs are inherently imperfect due2
PREPRINT to the modal truncation and intrinsic nonlinearity. We also perturb the initial conditions to further mimic erroneous stateestimates in practice. Meanwhile, we realize that, more often than not, sensor signals are noisy. So, we intentionallyinject some noise to the synthesized observation data (using a twin-experiment approach). We test the performance ofLSTM-N with various levels of measurement noises, initial field perturbations, and sensors signals sparsity.
Every aircraft generates a wake of turbulent air as it flies. This disturbance is caused by a pair of tornado-like counter-rotating vortices (called wake vortex) that trail from the tips of the wings [66]. Relatively turbulent weather conditionsand rough terrain can help dissipate these vortices. A faster wake-decay was seen with increase in terrain roughness(as in [67]), where it was observed that due to higher shear generated by the rougher terrain, a secondary vortex (SV)gets established more rapidly around the periphery of primary wake-vortex (WV), and the subsequent interactionsbetween SV and WV creates a higher turbulence state which destroys the associated vorticity and both these vortex.The phenomena like WV rebound and generation of omega-shaped hair-pin vortices also takes place during this SV-WVinteraction. This complex wake decay phenomena is also applicable for wake-vortex emanating from aircraft. Suchfacts have also been observed and exploited to artificially destroy wakes close to the ground using plates [68]. Therefore,understanding the complex dynamics of these wake vortices (WV) from its generation to decay is important in order toensure flight safety, to increase airport capacity and to test new methods for destroying WVs and mitigating their effect.Air traffic control can potentially benefit from the emerging concept of a digital twin (DT). DT is the virtual replica of aphysical system, where both are able to actively communicate with each other [69–71]. Given the WV and associatedairport traffic case, a DT would receive meteorological data concerning present as well as possible weather conditions.Inputs should also include airport traffic status, leading as well as training aircraft characteristics (e.g., weight, size) andflight mode (e.g., take-off or landing). Topography and geographic location can be a factor, too. Then, the DT shouldprocess this stream with data and assess a bunch of possible scenarios corresponding to different potential choices.Based on those assessments, an informed decision can be made with regard to separation distance and flight schedulingfor instance. Considering the WV problem, numerical modeling based on the Navier-Stokes equation, if accurate,can be a cost effective and easily employable pursuit for wake analysis. In Figure 1, an aircraft is admitted to landsafely, based on the decay of wake-vortices from a leading aircraft. These wake-vortices are generated using the aircraftinformation and an analytical function on a set of two-dimensional (2D) planes (also called gates) perpendicular tothe flight path [72]. An alternative and a more realistic initialization will be the one using LIDAR data, if availablein real time. The initialized wake vortices then decay and get transported under the influence of a background windand turbulence field. Once the flight corridor is clear and free of any influence of the wake-vortices left behind by theleading aircraft, the following aircraft can land or take-off safely.Figure 1: Transported and diffused wakes on a set of 2D planes (a.k.a. gates) to make sure that the flight corridor isclear for the following aircraft.Direct, full order numerical simulations require large discretized systems for adequate approximation and are notpractical for real-time wake prediction, which is an essential ingredient for feasible DT technologies. Therefore, reducedorder modeling (ROM) rises as a natural choice for successful implementation of digital twin applications. ROMrepresents a family of protocols that aim at emulating the relevant system’s dynamics with minimal computational3
PREPRINT burden. Typical ROM approaches usually consist of two major steps; (1) tailor a low-order subspace, where the flowtrajectory can be sufficiently approximated to live (see Section 3.2), (2) build a surrogate model to cheaply evolve thistrajectory in time (see Section 3.3). Traditionally, building surrogate models to evolve on a reduced manifolds hasrelied on the projection of the full order model (FOM) operators onto a reduced subspace (e.g., using Galerkin-typetechniques) to structure a reduced order model (ROM). Those FOM operators are usually the outcome of the numericaldiscretization of the well-established governing equation, derived from first principles and conservation laws. SuchROMs are attractive due to their reasonable interpretability and generalizability, as well as the existence of robusttechniques for stability and uncertainty analysis. However, Galerkin ROM (GROM) can be expensive to solve forturbulent and advection-dominated flows. GROM also might suffer from inaccuracies and instabilities for long-timepredictions. Meanwhile, in the digital twin context, the availability of rich stream of data and measurements opens newavenues for further ROM development. One way to utilize this abundance of data is the purely data-driven nonintrusiveROM (NIROM) approach. NIROMs have largely benefited from the widespread of open-source cutting edge andeasy-to-use machine learning (ML) libraries, and cheap computational infrastructure to solely rely on data in order tobuild stable and accurate models, compared to their GROM counterparts [73–83]. However, purely data-driven toolsoften lack human interpretability and generalizability, and sometimes become prohibitively “data-hungry”.Alternatively, hybrid approaches can be pursued, where data-driven tools only assist the physics-based models wheneverdata are available, rather than replacing them entirely. Data assimilation (DA) is a framework which can efficientlyachieve this objective. DA generally refers to the discipline of intelligently fusing theory and observations to yield anoptimal estimate of the system’s evolution [84–88]. Measurements are usually sparse (both in time and space) andnoisy, while dynamical models are often imperfect due to the underlying assumptions and approximations introducedduring either model derivation (e.g., neglecting minor source terms) or numerical solution of the resulting model (e.g.,truncation error). DA algorithms have rich history in numerical weather predictions and are utilized on a daily basis toprovide reliable forecasts. In this paper, we suppose that our dynamical model is the truncated GROM and we aimat utilizing live measurements to correct the GROM trajectory. Specifically, we exploit the nudging method as ourdata assimilation framework, which works by relaxing the model state toward observations by adding correction (ornudging) terms, to mitigate the discrepancy between observations and model state [89]. We employ LSTM mappings toaccount for this nudging term based on a combination between GROM predictions and available measurement data (seeSection 3.4).
In this section, we first give an overview of the full order model used to simulate the wake-vortex transport problem.Then, we present the reduced order formulations adopted in this study. In particular, we utilize proper orthogonaldecomposition (POD) as a data-driven tool to extract the flow’s coherent structures and build a reduced order subspacethat best approximate the flow fields of interest. Then, we perform a Galerkin approach to project the full order modeloperators onto that reduced space to build a “physics-constrained” reduced order model.
Here, we consider the two-dimensional (2D) vorticity transport equation as our full order model (FOM) that resolvesthe wake-vortex transport and decay. It refers to the 2D Navier-Stokes equations in vorticity-streamfunction formulationas follows [90], ∂ω∂t + J ( ω, ψ ) = 1 Re ∇ ω, (1)where ω and ψ denote the vorticity and streamfunction fields, respectively. Re is the dimensionless Reynolds number,defined as the ratio of inertial effects to viscous effects. Equation 1 is complemented by the kinematic relationshipbetween vorticity and streamfunction defined as, ∇ ψ = − ω. (2)Equation 1 and Equation 2 include two operators, the Jacobian ( J ( · , · ) ) and the Laplacian ( ∇ ( · ) ) defined as J ( ω, ψ ) = ∂ω∂x ∂ψ∂y − ∂ω∂y ∂ψ∂x , (3) ∇ ω = ∂ ω∂x + ∂ ω∂y . (4)In order to mimic the wake-vortex problem, several models have been investigated [1, 7, 91, 92], including Gaussianvortex [93], Rankine vortex [94, 95], Lamb-Oseen vortex [96, 97], Proctor vortex [98, 99], etc. In the present study,4 PREPRINT we initialize the flow with a pair of counter-rotating Gaussian vortices with equal strengths centered at ( x , y ) and ( x , y ) as follows, ω ( x, y,
0) = exp (cid:0) − ρ (cid:2) ( x − x ) + ( y − y ) (cid:3)(cid:1) − exp (cid:0) − ρ (cid:2) ( x − x ) + ( y − y ) (cid:3)(cid:1) , (5)where ρ is an interacting parameter that controls the mutual interactions between the two vortical motions. The first step for building a projection-based reduced order model is to tailor a low-order subspace that is capableof capturing the essential features of the system of interest. In the fluid mechanics community, proper orthogonaldecomposition (POD) is one of the most popular techniques in this regard [100–102]. Starting from a collectionof system’s realizations (called snapshots), POD provides a systematic algorithm to construct a set of orthonormalbasis functions (called POD modes) that best describes that collection of snapshot data (in the (cid:96) sense). Moreimportantly, those bases are sorted based on their contributions to the system’s total energy, making the modal selectiona straightforward process. This is a significant advantage compared to other modal decomposition techniques likedynamic mode decomposition, where further sorting and selection criterion has to be carefully defined. Usually, themethod of snapshots [103] is followed to perform POD efficiently and economically, especially for high dimensionalsystems. However, we demonstrate the singular value decomposition (SVD) based approach here for the sake ofsimplicity and brevity of presentation.Suppose we have a collection of N system realizations, denoted as ω k = { ω ( x i , y j , t k ) } i = N x ,j = N y ,k = Ni =1 ,j =1 ,k =1 , we builda snapshot matrix A ∈ R M × N as A = [ Ω , Ω , . . . , Ω N ] , where Ω k ∈ R M × is the k th snapshot reshaped into acolumn vector, M is the number of spatial locations (i.e., M = N x N y ) and N is the number of snapshots.Then, a thin (reduced) SVD is performed on A , A = UΣV T , (6)where U ∈ R M × N is a matrix with orthonormal columns are the left singular vectors of A , which represent the spatialbasis, while the columns of V ∈ R N × N are the right singular vectors of A , representing the temporal basis. Thesingular values of A are stored in descending order as the entries of the diagonal matrix Σ ∈ R N × N . For dimensionalityreduction purposes, only the first R columns of U , the first R columns of V , and the upper-left R × R sub-matrix of Σ are considered, corresponding to the largest R singular values. Specifically, the first R columns of U represent the mosteffective R POD modes, denoted as { φ k } Rk =1 for now on.The vorticity field ω ( x, y, t ) is thus approximated as a linear superposition of the contributions of the first R modes,which can be mathematically expressed as ω ( x, y, t ) = R (cid:88) k =1 a k ( t ) φ k ( x, y ) , (7)where φ k ( x, y ) are the spatial modes, a k ( t ) are the time-dependent modal coefficients (also known as generalizedcoordinates), and R is the number of retained modes in ROM approximation (i.e., ROM dimension). We note that thePOD basis functions φ are orthonormal by construction as (cid:104) φ i ; φ j (cid:105) = (cid:26) if i = j otherwise, (8)where the angle parentheses (cid:104)· ; ·(cid:105) stands for the standard inner product in Euclidean space (i.e., dot product).Meanwhile, since vorticity and streamfunction fields are related by Eq. 2, they can share the same modal coefficients( a k ( t ) ). Moreover, the basis functions for the streamfunction (denoted as θ k ( x, y ) ) can be derived from those of thevorticity as follows, ∇ θ k = − φ k , k = 1 , , . . . , R, (9)and the ROM approximation of the streamfunction can be written as ψ ( x, y, t ) = R (cid:88) k =1 a k ( t ) θ k ( x, y ) , (10)5 PREPRINT
After constructing a set of POD basis functions, an orthogonal Galerkin projection can be performed to obtain theGalerkin ROM (GROM). To do so, the ROM approximation (Eq. 7-10) is substituted into the governing equation(Eq. 1). Noting that the POD bases are only spatial functions (i.e., independent of time) and the modal coefficients areindependent of space, we get the the following set of ordinary differential equations (ODEs) representing the tensorialGROM d a k d t = R (cid:88) i =1 L i,k a i + R (cid:88) i =1 R (cid:88) j =1 N i,j,k a i a j , (11)where L and N are the matrix and tensor of predetermined model coefficients corresponding to linear and nonlinearterms, respectively. Those are precomputed during an offline stage as L i,k = (cid:10) Re ∇ φ i ; φ k (cid:11) , N i,j,k = (cid:10) − J ( φ i , θ j ); φ k (cid:11) . Equation 11 can be rewritten in compact form as ˙ a = f ( a ) , (12)where the (continuous-time) model map f is defined as follows, f = (cid:80) Ri =1 L i, a i + (cid:80) Ri =1 (cid:80) Rj =1 N i,j, a i a j (cid:80) Ri =1 L i, a i + (cid:80) Ri =1 (cid:80) Rj =1 N i,j, a i a j ... (cid:80) Ri =1 L i,R a i + (cid:80) Ri =1 (cid:80) Rj =1 N i,j,R a i a j . Alternatively, Eq. 12 can be given in discrete-time form as a n +1 = M ( a n ) , (13)where M ( · ) is the discrete-time map obtained by any suitable temporal integration technique. Here, we use thefourth-order Runge-Kutta (RK4) method as follows, a n +1 = a n + ∆ t g + 2 g + 2 g + g ) , (14)where g = f ( a n ) , g = f ( a n + ∆ t · g ) , g = f ( a n + ∆ t · g ) , g = f ( a n + ∆ t · g ) . Thus the discrete-time map defining the transition from time t n to time t n +1 is written as M ( a n ) = a n + ∆ t g + 2 g + 2 g + g ) . (15) Due to the quadratic nonlinearity in the governing equation (and consequently the GROM), the online computationalcost of solving Eq. 11 is O ( R ) (i.e., it scales cubically with the number of retained modes). Therefore, this has tobe kept as low as possible for feasible implementation of ROM in digital twin applications that require near real-timeresponses. However, this is often not an easy task for systems with slow decay of the Kolmogorov n-width. Examplesincludes advection-dominated flows with strong nonlinear interactions among a wide range of modes. As a result, theresulting GROM is intrinsically imperfect model. That is even with the true initial conditions, and absence of numericalerrors, the GROM might give inaccurate or false predictions.Moreover, in most realistic cases, proper specification of the initial state, boundary conditions, and/or model parametersis rarely attainable. This uncertainty in problem definition, in conjunction with model imperfection, poses challenges6 PREPRINT for accurate predictions. In this study, we put forth a nudging-based methodology that fuses prior model forecast (usingimperfect initial condition specification and imperfect model) with the available Eulerian sensor measurements toprovide more accurate posterior prediction. Relating our setting to realistic applications, we build our framework onthe assumption that measurements are noisy and sparse both in space and time. Nudging has a prestigious history indata assimilation, being a simple and unbiased approach. The idea behind nudging is to penalize the dynamical modelevolution with the discrepancy between model’s predictions and observations [104–106]. In other words, the forwardmodel given in Eq. 13 is supplied with a nudging (or correction) term rewritten in the following form, a n +1 = M ( a n ) + G ( z n +1 − h ( a n +1 )) , (16)where G is called the nudging (gain) matrix and z is the set of measurements (observations), while h ( · ) is a mappingfrom model space to observation space. For example, h ( · ) can be a reconstruction map, from ROM space to FOM space.In other words, h ( a ) represents the “model forecast for the measured quantity”, while z is the “actual” observations.Despite the simplicity of Eq. 16, the specification/definition of the gain matrix G is not as simple [63, 107–109].In the proposed framework, we utilize a recurrent neural network, namely the long short-term memory (LSTM) variant,to define the nudging map. In particular, Eq. 16 implies that each component of a n +1 (i.e., a , a . . . , a R ) is correctedusing a linear superposition of the components of z n +1 − h ( a n +1 ) , weighted by the gain matrix. Here, we relax thislinearity assumption and generalize it to a possibly nonlinear mapping C ( a , z ) as, a n +1 = M ( a n ) + C ( a n +1 b , z n +1 ) , (17)where the map C ( a , z ) is learnt (or inferred) using an LSTM neural network, and a n +1 b is the prior model predictioncomputed using imperfect model and/or imperfect initial conditions (called background in data assimilation terminology),defined as a n +1 b = M ( a n ) . Thus, Eq. 17 can be rewritten as follows, a n +1 = a n +1 b + C ( a n +1 b , z n +1 ) . (18)In order to learn the map C ( a b , z ) , we consider the case with imperfect model, defective initial conditions, and noisyobservations. Moreover, we suppose sensors are sparse in space and measurement signals are sparse in time, too.Specifically, we use sensors located at a few equally-spaced grid points, but a generalization to off-grid or adaptivesensor placement is possible. Also, we assume sensors send measurement signals every τ time units. In order to mimicsensor measurements and noisy initial conditions, we run a twin experiment as follows,1. Solve the FOM equation (i.e., Eq. 1) and sample true field data ( ω true ( x, y, t n ) ) each τ time units. In otherwords, store ω true ( x, y, t n ) at t n ∈ { , τ, τ, . . . T } where T is the total (maximum) time and τ is the timewindow over which measurements are collected.2. Define erroneous initial field estimate as ω err ( x, y, t n ) = ω true ( x, y, t n )+ (cid:15) b , where t n ∈ { , τ, τ, . . . T − τ } . (cid:15) b stands for noise in initial state estimate, assumed as white Gaussian noise with zero mean and covariancematrix B (i.e., (cid:15) b ∼ N (0 , B ) ).3. Define sparse and noisy measurements as z = ω true ( x Obs , y
Obs , t n ) + (cid:15) m , for t n ∈ { τ, τ, . . . T } . Similarly, (cid:15) m stands for the measurements noise, assumed to be white Gaussian noise with zero mean and covariancematrix Q (i.e., (cid:15) m ∼ N (0 , Q ) ).For LSTM training data, we project the erroneous field estimates (from Step 2) onto the POD basis functions to get theerroneous POD modal coefficients (i.e., a err ( t n ) , for t n ∈ { , τ, τ, . . . T − τ } . Then, we integrate those erroneouscoefficients for τ time units to get the background prediction a b ( t n ) , for t n ∈ { τ, τ, . . . T } .Then, we train the LSTM using a b ( t n ) and z ( t n ) as inputs, and set the target as the correction ( a true ( t n ) − a b ( t n )) ,for t n ∈ { τ, τ, . . . T } . The true modal coefficients ( a true ) are obtained by projecting the true field data (from Step 1)onto the POD bases, where the projection is defined via the inner product as a k ( t ) = (cid:104) ω ( x, y, t ); φ k ( x, y ) (cid:105) . In order toenrich the training data set, Step 2 and Step 3 are repeated several times giving an ensemble of erroneous state estimatesand noisy measurements at every time instant of interest. Each member of those ensembles represents one trainingsample. This also assists the LSTM network to handle wider range of noise.We emphasize that the proposed LSTM-N approach not only cures model imperfection (i.e., provides model closure aswell as accounts for any missing physical processes), but also treats uncertainties in initial state estimates. This mightbe caused by the selection of inaccurate wake vortex model, or the idealizations embedded in this model compared toreality. Moreover, the field measurements (i.e., the nudging data) are assumed to be sparse and noisy to mimic real-lifesituations. 7 PREPRINT
In order to test and verify the proposed ideas, we consider a square 2D domain with a side length of π . Flow is initiatedusing a pair of Gaussian vortices as given in Eq. 5 centered at ( x , y ) = (cid:18) π , π (cid:19) and ( x , y ) = (cid:18) π , π (cid:19) withan interaction parameter of ρ = π . Results in this section are shown at Re = 1000 . For FOM simulations, a regularCartesian grid resolution of × is considered (i.e., ∆ x = ∆ y = 2 π/ ), with a time-step of . . Snapshotsof vorticity fields are collected every 100 time-steps for t ∈ [0 , , totalling 300 snapshots. The evolution of the wakevortex problem is depicted in Figure 2, demonstrating the convective and interactive mechanisms affecting the transportand development of the two vortices.Figure 2: Evolution of the FOM vorticity field for the wake vortex transport problem with a Reynolds number of .Flow is initiated at time t = 0 with a pair of Gaussian distributed vortices.For ROM computations, 6 modes are retained in the reduced order approximation (i.e., R = 6 ) and a time step of . isadopted for the temporal integration of GROM equations. In order to implement the LSTM-N approach, we begin aterroneous initial condition defined as ω err ( x, y,
0) = ω true ( x, y,
0) + (cid:15) b , where ω true ( x, y, is defined with Eq. 5,and (cid:15) b is a white Gaussian noise with zero mean and covariance matrix B . For simplicity, we assume B = σ b I , where σ b is the standard deviation in the “background” estimate of the initial condition and I is the identity matrix. We notethat this formulation implies that the errors in our estimates of the initial vorticity field at different spatial locations areuncorrelated. As nudging field data, we locate sensors to measure the vorticity field ω ( x, y, t ) every 64 grid points (i.e.,9 sensors in each direction, with s freq = 64 , where s freq is the number of spatial steps between sensors locations),and collect measurements every 10 time steps (i.e., each time unit with t freq = 10 , where t freq is the number oftime steps between measurement signals). To account for noisy observations, a white Gaussian noise of zero meanand covariance matrix of Q is added to the true vorticity field obtained from the FOM simulation at sensors locations.Similar to B , we set Q = σ m I , where σ m is the standard deviation of measurement noise. This assumes that the noisein sensors measurements are not correlated to each other, and all sensors have similar quality (i.e., add similar amountsof noise to the measurements). As a base case, we set σ b = 1 , and σ m = 0 . .The procedure presented in Sec. 3.4 is applied using the numerical setup described above, and compared againstthe reference case of GROM with the erroneous initial condition and inherent model imperfections due to modaltruncation (denoted as background forecast). In Fig. 3, the temporal evolution of the POD modal coefficients isshown for the true projection, background, and LSTM-Nudge results. The true projection results are obtained by theprojection of the true FOM field at different time instants onto the corresponding basis functions (i.e., via inner product, a k,true ( t ) = (cid:104) ω ( x, y, t ); φ k ( x, y ) (cid:105) ). The background trajectory is the reference solution obtained by standard GROMusing the erroneous initial condition, without any closure or corrections. It can be seen that the background trajectorygets off the true trajectory by time as a manifestation of model. Also, note that the background solution does notbegin from the same point as true projection due to the noise in the initial condition. On the other hand, the LSTM-Npredictions almost perfectly match the true projection solution, implying that the approach is capable of blending noisyobservations with a prior estimate to gain more accurate predictions.In order to better visualize the predictive capabilities of the LSTM-N methodology, we compute the reconstructedvorticity field using Eq. 7. The final field reconstruction (at t = 30 ) is shown in Figure 4, comparing the true projection,background, LSTM-N results. Note that the field obtained from true projection at any time instant can be computed as ω true ( x, y, t ) = (cid:80) Rk =1 a k,true ( t ) φ k ( x, y ) , and represents the optimal reduced-rank approximation that can be obtained8 PREPRINT
Figure 3: Temporal evolution of the POD modal coefficients for the 2D wake vortex transport problem. [Base case with σ b = 1 and σ m = 0 . ]using a linear subspace spanned by R bases. Comparing true projection results from Figure 4 against FOM at finaltime from Figure 2 reveals that, for this particular case, 6 modes are qualitatively capable to capture most of therelevant features of the flow field. The LSTM-N outputs significantly match the projection of the FOM field, while thebackground forecasts show some visible deviations.Figure 4: Final vorticity field (at t = 25 ) for the wake-vortex transport problem, with σ b = 1 . , and σ m = 0 . .For further quantitative assessment, the root mean-squares error RM SE of the reconstructed field with respect to theFOM solution is calculated as a function of time as follows,
RM SE ( t ) = (cid:118)(cid:117)(cid:117)(cid:116) M N x (cid:88) i =1 N y (cid:88) j =1 (cid:18) ω F OM ( x i , y j , t ) − ω ROM ( x i , y j , t ) (cid:19) , (19)where ω F OM is the true vorticity field obtained from solving the FOM equation, while ω ROM is the reduced orderapproximation computed through true projection, background (reference) solution, or LSTM-N method. The
RM SE at different times is plotted in Figure 5, demonstrating the capability of LSTM-N framework to efficiently recover theoptimal reconstruction given a few sparse measurements.
Next, We explore the effect of noise on the LSTM-N results. In other words, we investigate how much noise theframework can tolerate. We note that we keep the same LSTM, trained with the base level of noise (i.e., σ b = 1 . and σ m = 0 . ) while we test it using different levels of noise. First, we gradually increase the standard deviation ofmeasurement noise from . to . (5 times larger) and . (10 times larger). In Figure 6, we plot the temporal RM SE metrics as well as field reconstruction at final time. We find that performance deteriorates a bit with that9
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Figure 5: Root mean-squares error for the wake-vortex transport problem, with σ b = 1 . , and σ m = 0 . .increase in measurement noise although results are still significantly better than the background forecast (starting fromthe same initial conditions). (a) σ m = 0 . , σ b = 1 . (b) σ m = 0 . , σ b = 1 . Figure 6: Reconstructed vorticity fields at final time, along with
RM SE for different levels of measurement noise.For testing the effect of initial state perturbation, we increase σ b from to , and . Figure 7 display the effect of thoselevels of initial field perturbations on background forecasts. Despite that, LSTM-N is performing very well even at thosehigh levels of initial peturbations. This is even clearer from the RM SE plots, beginning from relatively large values10
PREPRINT and quickly decaying to the level of true projection once measurements are available. From Figure 6 and Figure 7, wecan deduce that the influence of the level of measurement noise on LSTM-N performance is more prominent that of theinitial field perturbation. We reiterate that in both cases, the LSTM is trained with σ b = 1 . and σ m = 0 . and testedfor different values. (a) σ m = 0 . , σ b = 5 . (b) σ m = 0 . , σ b = 10 . Figure 7: Reconstructed vorticity fields at final time, along with
RM SE for different levels of background noise.
Finally, we consider the effect of measurement sparsity on the accuracy of the presented approach. This is crucialfor the trade-off between quality and quantity of sensors, since it has been shown in Section 4.1 that measurementnoise significantly affects the LSTM-N output. For the base case, sensors are placed at every grid points. Now,we place sensors every 32 grid points, representing a denser case, as well as and grid points, representingscarcer sensors. We find that the framework is quite robust, providing very good results as illustrated in Figure 8. Wenote here, however, that the same original LSTM cannot be utilized for testing with varying sparsity. This is becausesensors sparsity controls the size of the input vector. Therefore, a new LSTM has to be re-trained for each case with thecorresponding number of measurements. We also emphasize that compressed sensing techniques should be adopted foroptimized sensors placement, rather than the simple collocated equidistant arrangement followed in the present study.Regarding temporal sparsity, we collect measurement each , , and time-steps, compared to the reference casewhere measurement are collected every time-steps. We can see from Figure 9 that all cases yield very goodpredictions. Furthermore, RM SE plots provide valuable insights about the capability of LSTM-N to effectively fusemeasurement with background forecast to produce more accurate state estimates. For example, when measurementsignals are collected every time-steps, this corresponds to time-units, meaning that the LSTM-N directly adopts theGROM prediction without correction for this amount of time, before correction is added. This is evident from Figure 9c,where the red curve starts and continues with the orange curve, then a sharp reduction of the RM SE is observed. Thisbehavior is repeated as the red curve departs from the blue one (corresponding to true projection) before correction is11
PREPRINT (a) sensors every 32 grid points (b) sensors every 128 grid points (c) sensors every 256 grid points
Figure 8: Comparison of resulting vorticity fields at final time as well as the line plots of root mean-squares error withtime, with different number of sensors located sparsely at grid points.added every τ = 3 time-units (i.e., time-steps). On the other hand, when more frequent measurement signals areavailable (e.g., every time-steps), deviation from the true projection results is less observed, as shown in Figure 9a. We demonstrate hybrid analysis and modeling (HAM) as an enabler for digital twin application of an airport. Specifically,we investigate the problem of wake-vortex transport and decay as a key factor for the determination of separationdistance between consecutive aircraft. Reduced order modeling based on Galerkin projection and proper orthogonaldecomposition is adopted to provide computationally light models. We develop a methodology to exploit machinelearning to cure model deficiency through online measurement data adopting ideas from dynamic data assimilation.Specifically, an LSTM architecture is trained to nudge prior predictions toward optimal state values using a combinationof background information along with sparse and noisy observations. The proposed framework is distinguished fromprevious studies in the sense that it is built on the assumption that all the computing ingredients are intrinsicallyimperfect, including a truncated GROM model, erroneous initial conditions, and defective sensors.We study the effects of measurement noise and initial condition perturbation on LSTM-N behavior. The frameworkworks sufficiently well for a wide range of noise and perturbation. Nonetheless, numerical experiments indicaterelatively more dependence of performance on measurement quality (noise). Meanwhile, we find that sensors sparsityhas minimal effects on results. We emphasize that the proposed framework represents a way of merging humanknowledge, physics-based models, measurement information, and data-driven tools to maximize their benefits ratherthan discarding any of them. The presented framework paves the way for viable digital twin applications to enhanceairports capacities by regulating air traffic without compromising consecutive aircraft safety. Nonetheless, the scalabilityof the approach has yet to be tested using different vortex models and taking into account other effective factors (e.g.,wind). 12
PREPRINT (a) measurements every 5 time steps (b) measurements every 20 time steps (c) measurements every 30 time steps
Figure 9: Comparison of resulting vorticity fields at final time as well as the line plots of root mean-squares error withtime using different measurement signal frequencies.
Acknowledgments
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of AdvancedScientific Computing Research under Award Number de-sc0019290. O.S. gratefully acknowledges the U.S. DOE EarlyCareer Research Program support. The work of A.R. and M.T. was supported by funding from the EU SESAR (SingleEuropean Sky ATM Research) program.Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government.Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty,express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, orotherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United StatesGovernment or any agency thereof. The views and opinions of authors expressed herein do not necessarily state orreflect those of the United States Government or any agency thereof.
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