Neural Vortex Method: from Finite Lagrangian Particles to Infinite Dimensional Eulerian Dynamics
aa r X i v : . [ phy s i c s . c o m p - ph ] J un N EUR AL V ORTEX M ETHOD : FROM F INITE L AGR ANGIAN P ARTIC LES TO I NFINITE D IMENSIONAL E ULER IAN D YNAMICS
Shiying Xiong ∗ Department of Computer ScienceDartmouth CollegeHanover, NH 03755
Xingzhe He
Department of Computer ScienceDartmouth CollegeHanover, NH 03755
Yunjin Tong
Department of Computer ScienceDartmouth CollegeHanover, NH 03755
Bo Zhu
Department of Computer ScienceDartmouth CollegeHanover, NH 03755June 9, 2020 A BSTRACT
In the field of fluid numerical analysis, there has been a long-standing problem: lacking of a rigor-ous mathematical tool to map from a continuous flow field to discrete vortex particles, hurdling theLagrangian particles from inheriting the high resolution of a large-scale Eulerian solver. To tacklethis challenge, we propose a novel learning-based framework, the Neural Vortex Method (NVM),which builds a neural-network description of the Lagrangian vortex structures and their interactiondynamics to reconstruct the high-resolution Eulerian flow field in a physically-precise manner. Thekey components of our infrastructure consist of two networks: a vortex representation network toidentify the Lagrangian vortices from a grid-based velocity field and a vortex interaction networkto learn the underlying governing dynamics of these finite structures. By embedding these two net-works with a vorticity-to-velocity Poisson solver and training its parameters using the high-fidelitydata obtained from high-resolution direct numerical simulation, we can predict the accurate fluiddynamics on a precision level that was infeasible for all the previous conventional vortex methods(CVMs). To the best of our knowledge, our method is the first approach that can utilize motions offinite particles to learn infinite dimensional dynamic systems. We demonstrate the efficacy of ourmethod in generating highly accurate prediction results, with low computational cost, of the leapfrog-ging vortex rings system, the turbulence system, and the systems governed by Euler equations withdifferent external forces.
Accurately capturing and quantifying the motions of fluid with fine details has been a challenging task of fluid numeri-cal analysis for centuries. From the theoretical standpoint, one of the most critical sets of governing equations of fluiddynamics is the Euler equations. Given a fluid velocity field u ( x , t ) with an incompressible constraint, its underlyingdynamics can be described by ∂ u ∂t + ( u · ∇ ) u = − ρ ∇ p + f , ∇ · u = 0 , (1)where t denotes the time, p the pressure, ρ the density, and f the body accelerations (per unit mass) acting on thecontinuum, for example gravity, inertial accelerations, electric field acceleration, and so on. ∗ corresponding author, email: [email protected] PREPRINT - J
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9, 2020To computationally solve the Euler equations, many numerical methods are developed and among which the mostwidely used one is the conventional vortex methods (CVMs) [34]. CVMs enjoy several advantages such as its au-tomatic adaptivity of the computational elements, the numerical conservation of physically conserved quantities, theability to simulate phenomena covering many orders of magnitude, and the rigorous treatment of boundary conditionsat infinity. CVMs are based on the discretization of the vorticity field ω = ∇ × u and the Lagrangian description ofthe governing equations (1) that, when solved, determine the evolution of the computational elements. Specifically,discretizing the (1) with N particles results in a set of ordinary differential equations (ODEs) for the particle strengths Γ = { Γ i | i = 1 , · · · , N } and the particle positions X = { X i | i = 1 , · · · , N } as d Γ i d t = γ i ( Γ , X ) , d X i d t = u i + v i ( Γ , X ) . (2)Here, Γ i is the integral of ω over the i th computational element, γ i is the change rate of the particle strength due tothe vortex stretching, u i = u ( X i ) is the velocity at X i , and v i is a drift velocity caused by the vorticity distributionand the external force [12].However, the implementation of CVM faces a major challenge that is to model the right-hand sides (RHSs) of (2)based on (1). In the CVMs, the velocity u i = u ( X i ) on the RHSs of (2) is evaluated by the Biot–Savart law (BS law).If the volume of vortex element is not considered, u i can be calculated by u i = 12( n d − π N X j = i Γ j × ( X i − X j ) | X i − X j | n d , (3)where n d is the dimension of the flow field. However, there are several drawbacks of using BS law. First, theassumption that the vortices are point-like largely limits the use of BS law. Second, the drift velocity v i due to theexternal force cannot be obtained using CVM without knowing the function of the external force. Even given thefunction, CVM still fails to capture v i accurately in most cases. Lastly, when there are two particles that are closeenough, equation (3) will have a large error. All of the above problems make CVM inaccurate and inapplicable insolving the underlying fluid dynamics under many situations [7].An alternative method, grid method, is developed to solve the accuracy issue that arises in the vortex method. Manyresearchers are devoted to high-resolution large-scale Eulerian solver [25, 5]. However, an unignorable problem of thegrid method is its high computational cost that is sometimes unaffordable for realistic conditions.To quantify the fluid dynamics accurately in an efficient manner, we propose a novel framework, Neural Vortex Method(NVM), that extracts information from the Eulerian specification of the flow field (or the images of flow visualizations)and translates it into knowledge about the underlying fluid field through physics-informed neural networks. Learningdirectly from high-dimensional observations, such as images, is unable to be achieved using traditional methods,since extracting the velocity and pressure fields directly from the images is challenging. We address this problem byconstructing a vortex representation network (section 2.2 Detection Network) to identify the positions and the vorticityof Lagrangian vortices from a grid-based velocity field, which from a mathematical perspective connects (1) with (2).In this way, we simplify the vorticity field into a field only consists of the identified vortices. Given the detectedvortices, we then use a vortex interaction network (section 2.3 Dynamics Network) to learn the underlying governingdynamics of these finite structures. Dynamics networks model the RHSs of 2 accurately under a variety of conditions,resolving the long-standing problem in CVM. To build a fully automated tool-chain that can construct a high-resolutionEulerian flow field from the Lagrangian inductive priors, we embed these two networks with a vorticity-to-velocityPoisson solver and trained its parameters using the high-fidelity data obtained from high-resolution direct numericalsimulation. The model is trained only with information collected from the interaction of 2 to 6 vortices, and the trainedmodel can be applied to any arbitrary vorticity field with any number of vortices.We demonstrate the efficacy of our method in generating highly accurate prediction results, with low computationalcost, of the leapfrogging vortex rings system, the turbulence system, and the systems governed by Euler equationswith different external forces, that are challenging to model for CVMs. As a example, Figure 1 depicts the two-dimensional Lagrangian scalar fields at t = 1 with the initial condition φ = x and resolution . The evolution ofthe Lagrangian scalar fields are induced by O (10) and O (100) random NVM vortex particles. With a small numberof NVM vortex particles, shown in Figure 1 (a), the spiral structure [26, 27] of individual NVM vortex particles canbe observed clearly. With a large number of NVM vortex particles, the underlying field exhibits turbulent behaviors.Generally, the high-resolution results shown in Figure 1 can only be achieved by supercomputation using grid-basedmethods [46], while NVM allows these to be generated on any laptop with GPU. Although CVM does not require sucha large computational cost as grid methods, suffering from inaccuracy, it can never produce such an accurate depictionof Lagrangian fields. 2 PREPRINT - J
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Figure 1:
Two-dimensional Lagrangian scalar fields at t = 1 with the initial condition φ = x and resolution .The evolution of the Lagrangian scalar fields are induced by (a) O (10) and (b) O (100) random NVM vortex particles.Using NVM as an alternative of the traditional vortex particle methods and grid-based methods has several benefits: Accuracy
NVM can predict the accurate fluid dynamics on a precision level that was infeasible for all the previousCVMs.
Efficiency
Our method only requires training data collected from a very short window of training period (more than100 times shorter than the prediction period). Compared with the grid-based methods, NVM can achieve the samelevel of accuracy with much lower computational cost.
Feasibility
NVM can be applied to fluid fields with any number of vortices with or wihtout volume and with anyboundary condition. It can capture the dynamics of the fluid under the influence of external forces. Moreover, dueto the short training period NVM requires, NVM can be applied to predict real systems that are hard to acquire datafrom.
Adaptivity
NVM inherits the ability of automatically adapting the computational elements from CVM.
The training process of NVM consists of 3 major steps: dataset generation, detection network training, and dynamicsnetwork training. we utilize data collected from randomly generated vortices and the corresponding vorticity fields totrain the detection network. The direct numerical simulation (DNS) is used to calculate the evolution of the vorticityfields. We use the well-trained detection network to identify the positions and the vorticity of vortices from the initialand the evolved vorticity fields. This process is shown in Figure 2. The identified vortices are then used to train thedynamics network.To predict the future flow, we use the trained detection network to simplify the vorticity field into a field only consistsof the identified vortices. Given the detected vortices, we then use the dynamics network to learn the underlyinggoverning dynamics of these finite structures.
We randomly sample 2 to 6 vortices and create the initial vorticity field though convolution with a Gaussian kernel ∼ N (0 , . . This process is repeated for 2000 times to generate 2000 samples. DNS is performed to solve (1) in theperiodic box using a standard pseudo-spectral method [32]. Aliasing errors are removed using the two-thirds truncationmethod with the maximum wavenumber k max ≈ N/ . The Fourier coefficients of the velocity is advanced in time3 PREPRINT - J
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Figure 2:
A example of vorticity contour at (a) t = 0 , (b) t = 0 . , and (c) superposition of t = 0 and t = 0 . . Theblack circles indicate the location recognized by the detection network. The evolution from (a) to (b) is calculated byDNS. Figure 3:
The architecture of the detection. It takes the vorticity field as input and output the position and vortexvolume for each vortex detected. The
Conv means the Conv2d-BatchNorm-ReLU combo and the
ResBlock is thesame as in [13]. In each
ResBlock , we use stride 2 to downsample the feature map. The number in the parenthesis isthe output dimension.using a second-order Adams–Bashforth method, and the time step is chosen to ensure that the Courant–Friedrichs–Lewy number is less than . for numerical stability and accuracy. The pseudo-spectral method used in this DNS issimilar to that described in [41, 42, 43, 44]. The input of the detection networks is a vorticity field of size × × . As shown in Figure 3, we first feed thevorticity field into a small one-stage detection network and get the feature map of size × × (we downsampled3 times). The primary reason for downsampling is to avoid extremely unbalanced data and multiple prediction for thesame vortex. We then forward the feature map to 2 branches. In the first branch, we conduct a × convolution togenerate a probability score ˆ p of the possibility that there exists a vortex. If ˆ p > . , we believe there exists a vortexwithin the corresponding cells of the original × × vorticity field. In the second branch, we predict the relativeposition to the left-up corner of the cell of the feature map if the cell contains a vortex. Afterwards, we set a boundingbox of × around these predicted vortices and use weighted average of the positions of the cells of the originalvorticity field to find a exact position of the vortex. Finally, the vortex volume is calculated as the sum of the value ofthe cells in the bounding box normalized by the cell area.Note that in the training process, we penalize the wrong position detection only if the cell that contains a vortex in theground truth given by DNS is not detected. This idea is similar to [31]. We do not use the weighted average method tofind the position in the training to ensure the detection network can produce detection results as accurate as possible.We use the focal loss [23] to further relief the unbalanced classification problem.The main reason we use the detection network to generate training data for the dynamics network is that we want touse the high-resolution data generated by the method mentioned in 2.1 instead of by the approximate particle method(BS law). Moreover, there are many situations where BS law is inapplicable, discussed previously in the section 1.The detection network enables us to find the positions of the vortices accurately regardless of the situations.4 PREPRINT - J
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Figure 4:
The architecture of the dynamics network. It takes the particles attribution as input and output the positionfor each vortex. The
ResBlock has the same architecture as in [13] with the convolution layers replaced by linear layers.The number in the parenthesis is the output dimension.The detection network is responsible for providing necessary information to the dynamics network. After the trainingprocess, we use the well-trained detection network to detect the vortices in the initial vorticity fields and the evolvedvorticity field, both generated by the method in section 2.1. We then apply the nearest-neighbor method to pair thevortices detected in these two fields. Figure 2 shows the case of two fields at t = 0 and t = 0 . . The idea ofnearest-neighbor pairing can be perceived from Figure 2 (c). The sample, or these two fields, is dropped if differentnumbers of vortices are detected in the initial and evolved fields or there exists large difference in the vorticity ofpaired vortices. The successfully detected vortices in the initial and evolved vorticity fields are passed together intothe dynamics network for its training. To learn the underlying dynamics of the vortices, we build a graph neural network similar to [2]. We predict thevelocity of one vortex due to influences exerted by the other vortices and the external force, and use the fourth-orderRunge–Kutta integrator to calculate the position in the next timestamp. As shown in Figure 4, for each vortex, we usea neural network A ( θ ) to predict the influences exerted by the other vortices and add them up. Specifically, for eachvortex i , we consider the vortex j ( j = i ) . The difference of their positions can be calculated by diff ij = pos i − pos j ,and their L2 distance is dist ij = k diff ij k . The input of the A ( θ ) is the vector ( diff ij , dist ij , vort j ) of length 4. Here,pos and vort are detected by the detection network. The output is the influence of the vortex j on the vortex i . In thisway, we can calculate the influence of each vortex j ( j = i ) on the vortex i . We sum up the all the influences on thevortex i and treat the result as the influence exerted by the other vortices.In addition, we use another neural network A ( θ ) to predict the global influence caused by the external force, whichis determined by the vorticity and the position of the vortex. The input of A ( θ ) is a vector of length 3. The output isthe influence exerted by the environment on the vortex i . Note that both the outputs of A ( θ ) and A ( θ ) are of length2. Thus, we can add the two kinds of influence together, whose result is defined as the velocity of the vortex i . Wefeed the velocity into the fourth-order Runge-Kutta integrator to obtain the predicted position of vortex i . We assumethe value of the vorticity is constant. In practice, the length of each time step used in the fourth-order Runge–Kuttaintegrator is 0.1s, and the total length is 0.2s. For both the detection network and the dynamics network, we use Adam optimizer [21] with learning rate 1e-3. Thelearning rate decays every 20 epochs by a multiplicative factor of 0.8. For the detection network, we use a batch size of32 and train it for 350 epochs. We use the cross entropy as the classification loss and use L1 loss for position prediction.To relief the unbalanced data problem in the detection network, we implement Focal loss [23] with α = 0 . and γ = 2 .It takes 15 minutes to converge on a single Nvidia RTX 2080Ti GPU. For the dynamics network, we use a batch sizeof 64 and train it for 500 epochs. We use L1 loss for the position prediction. It takes 25 minutes to converge on asingle Nvidia RTX 2080Ti GPU. 5 PREPRINT - J
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Figure 5:
Comparison of NVM and CVM for solving Euler equation in the periodic box. (a) NVM, (b) CVM, and(c) The relative error of velocity in flow simulation. The red dots indicate the positions of 2 vortices at different timesteps generated by DNS, and the black-circles in (a) and (b) are the prediction results of NVM and CVM, respectively.The black arrows indicate the directions of the motions of the 2 vortices.
To demonstrate that NVM is a better approach to capture the fluid dynamics than the traditional methods, we comparethe prediction results made by NVM and CVM for solving Euler equations in the periodic box. We plot the resultsusing NVM and CVM, and the relative error of velocity in the simulation in Figure 5 (a), (b), and (c) respectively.The red dots indicate the positions of 2 vortices at different time steps generated by DNS, and the black circles are theprediction results of NVM and CVM. It is quite obvious that in Figure 5 (a) the predictions made by NVM match thepositions of vortices generated by DNS almost perfectly, while the predictions made by BS law in Figure 5 (b) containa large error. The divergence of the relative error of velocity shown in Figure 5 (c) as t increases also shows that NVMoutperformances the traditional methods by an increasing amount as the predicting period becomes longer. A classic example of interesting filament dynamics are the leapfrogging vortex rings, which is an axisymmetric laminarflow. This phenomenon is typically very hard to reproduce in standard fluid solver [6], especially to keep the symmetricstructure. Here, we use NVM to predict the motions of 4 vortices and add 80000 randomly initialized tracers for bettervisualization. Since the tracers do not affect the dynamics of the underlying vorticity field, we use BS law to calculatethe motions of these tracers for faster visualization. Figure 6 shows the evolution of the vorticity field predicted byNVM under the initial condition of leapfrogging vortex rings at t = 0 , t = 11 , t = 22 , and t = 33 . NVM accuratelycaptures the symmetric structure of the leapfrogging vortex rings without losing such feature as time evolves.Besides simple systems like leapfrogging vortex rings, NVM is capable of predicting complicated turbulence systems.Figure 1 depicts the two-dimensional Lagrangian scalar fields at t = 1 with the initial condition φ = x and resolution . The governing equation of the Lagrangian scalar fields is ∂φ∂t + u · ∇ φ = 0 . (4)The evolution of the Lagrangian scalar fields are induced by O (10) and O (100) random NVM vortex particles. Basedon the particle velocity field from the NVM, a backward-particle-tracking method is applied to solve (4), and thenthe iso-contour of the Lagrangian field can be extracted as material structures in the evolution [46, 45, 47, 49, 48].In Figure 1 (a), the spiral structure [26, 27] of individual NVM vortex particles can be observed clearly due to thesmall number of NVM vortex particles. In Figure 1 (b), the underlying field exhibits turbulent behaviors, since itis generated with a large number of NVM vortex particles. We demonstrate that NVM is capable of generating anaccurate depiction of complex turbulence systems with low computational cost. In Figure 7, we show NVM’s ability of stably making accurate predictions of fluid dynamics governed by Eulerequations with different external forces, which are (a) f = , (b) f = 0 . ω (1 , , and (c) f = 0 . ω (cos( x − x c ) , − sin( y − y c )) . ω represents the vorticity, and ( x c , y c ) is the center of the computation domain. Here, we did not6 PREPRINT - J
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Figure 6:
Vorticity field predicted by 4 NVM vortices under the initial condition of leapfrogging vortex rings at t = 0 , t = 11 , t = 22 , and t = 33 . Vortices are indicated by the white–black circles. For better visualization, we use 80000tracers. Figure 7:
Prediction results using NVM for three cases with (a) f = , (b) f = 0 . ω (1 , , and (c) f =0 . ω (cos( x − x c ) , − sin( y − y c )) . The black arrows indicate the directions of the motions of the 2 vortices. ω represents the vorticity, and ( x c , y c ) is the center of the computation domain.plot the results generated by CVM, because CVM is incapable of solving the fluid dynamics with non-zero externalforces, since it fails to capture the drift velocity v i caused by the external force. Computational fluid dynamics
In the field of computational fluid dynamics (CFD), numerical analysis and datastructures are used to analyze and solve the motions of fluid flows. Two general approaches are developed to achievethis task, one is grid-based, or Eulerian, and the other is particle-based, or Lagrangian. Conventionally, many piecesof research are done on grid-based methods mainly to achieve high resolution, for example on channel flow [20, 50],boundary layers [40, 33], isotropic turbulence [16, 18, 17], and pipe flow [39]. However, the main drawback ofgrid-based methods is their high computational cost that is unaffordable under many conditions. On the contrary, theparticle-based approach is efficient in computation but suffers from inaccuracy in results [24, 25]. CVMs are well-known particle-based methods, which use vortices as the computational elements, mimicking the physical structures7
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9, 2020in turbulence. In CVMs, the Euler equations are formulated in terms of vorticity in contrast to the conventional for-mulations with velocity and pressure and are solved using a Lagrangian approach instead of the Eulerian formulation[22, 7, 1].
Machine learning in fluid systems
The rapid advent of machine learning techniques is opening up new possibilitiesto solve the physical system’s identification problems by statistically exploring the underlying structure of a varietyof physical systems, encompassing applications in quantum physics [35], thermodynamics [14], material science [36],rigid body control [9], Lagrangian systems [8], and Hamiltonian systems [10, 19, 37]. Specifically, in the field of fluidmechanics, machine learning offers a wealth of techniques to extract information from data that could be translatedinto knowledge about the underlying fluid field, as well as exhibits its ability to augment domain knowledge andautomate tasks related to flow control and optimization [4, 11]. Recently, many pieces of research are developed toefficiently learn the fluid dynamics through incorporating physical priors into the learning framework, e.g., encodingthe Navier-Stokes equations [29], embedding the notion of an incompressible fluid [28], and identifying a mappingbetween the dynamics of wave-based physical phenomena and the computation in a recurrent neural network (RNN)[15].
Learning physics laws from high dimensional observations
One of the key strengths of neural networks is that theycan learn abstract representations directly from high-dimensional data such as pixels. With the advances of imagedetection techniques [13, 31], it is natural to apply these techniques to better learn and predict physical phenomena.Belbute-Peres et al., attempted to learn a model of physical system dynamics end-to-end from image sequences usingan autoencoder [3]. Greydanus et al., also tried to combine an autoencoder with an Hamiltonian Neural Networks(HNN) to model the dynamics of pixel observations of a pendulum [10]. Toth et al., later developed HamiltonianGenerative Network (HGN), which is capable of consistently learning Hamiltonian dynamics from high-dimensionalobservations without restrictive domain assumptions [38]. Some related works done on fluid dynamics include HiddenFluid Mechanics (HFM) by Raissi et al., [30]. However, HFM differs from our work by learning the underlyinggoverning function purely based on the Eulerian grid.
This paper presented NVM, a novel learning-based framework, which builds a neural-network description of theLagrangian vortex structures and their interaction dynamics to reconstruct the high-resolution Eulerian flow field ina physically-precise manner. We demonstrated the efficacy of our method in generating highly accurate predictionresults, with low computational cost, of the leapfrogging vortex rings system, the turbulence system, and the systemsgoverned by Euler equations with different external forces. We compared the prediction results made by NVM andCVM for solving Euler equation in the periodic box and discovered that the relative error of predict velocity usingNVM is more than 10 times smaller than that of CVM. Moreover, our method only requires training data collected froma very short window of training period (more than 100 times shorter than the prediction period), which will potentiallysolve data acquisition problems in real systems. Our method is the first approach that can utilize motions of finiteparticles to learn infinite dimensional dynamic systems. Featured by its unique ability to generate highly accurateprediction results with low computational, NVM marks a significant advancement in numerical fluid simulation.
Accurately capturing and quantifying the motions of fluid has been a long-standing subject in the study of fluid dynam-ics, embracing a broad application in fields, such as fluid mechanics, wave physics, thermodynamics, quantum physics,aerodynamic, geophysics, electromagnetics, engineering and biomedicine. Our method, highlighted by its ability togenerate highly accurate prediction results with low computational cost, marks a critical improvement in numericalfluid simulation. Moreover, our method only requires training data collected from a very short window of trainingperiod (more than 100 times shorter than the prediction period), which will potentially solve data acquisition problemsin real systems. This research does not bring any direct ethical consequence, but the application of our method to fieldslike biomedicine can potentially cause ethical issues.
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