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2021,  3 (2):   87 - 104   Published Date:2021-4-20

DOI: 10.1016/j.vrih.2021.02.002
1 Introduction2 Physical-based fluid simulation method2.1 Eulerian method2.2 Lagrangian method2.3 Hybrid method3 Data-driven methodology3.1 Eulerian data-driven methods3.2 Lagrangian data-driven methods3.3 Hybrid data-driven methods4 Application4.1 Data-driven solver4.2 Detail enhancement4.3 Animation synthesis4.4 Fluid control4.5 Differentiable simulation5 Technical issues and future directions5.1 Interpretability5.2 Physical fidelity5.3 PDE solving5.4 Performance optimization5.5 Generalization5.6 Model reduction5.7 Data processing5.8 Dataset framework6 Conclusion


The field of fluid simulation is developing rapidly, and data-driven methods provide many frameworks and techniques for fluid simulation. This paper presents a survey of data-driven methods used in fluid simulation in computer graphics in recent years. First, we provide a brief introduction of physical-based fluid simulation methods based on their spatial discretization, including Lagrangian, Eulerian, and hybrid methods. The characteristics of these underlying structures and their inherent connection with data-driven methodologies are then analyzed. Subsequently, we review studies pertaining to a wide range of applications, including data-driven solvers, detail enhancement, animation synthesis, fluid control, and differentiable simulation. Finally, we discuss some related issues and potential directions in data-driven fluid simulation. We conclude that the fluid simulation combined with data-driven methods has some advantages, such as higher simulation efficiency, rich details and different pattern styles, compared with traditional methods under the same parameters. It can be seen that the data-driven fluid simulation is feasible and has broad prospects.


1 Introduction
Fluid simulations have been applied widely in computer graphics. Instead of discrete models, including the particle Bhatnagar-Gross-Krook[1], particle Fokker-Planck[2], direct simulation Monte Carlo-computational fluid dynamics (CFD) hybrid[3], multiscale particle[3], and coarse-grained particle simulation methods[4], we focus primarily on a continuum model governed by the well-known Navier-Stokes (N-S) equation that is the mainstream model in computer graphics. To simplify the physical model, the mass conservation of a constant density field is typically enforced under the incompressibility assumption for water and smoke animation. In addition, other physical forces, such as surface tension, viscosity, and vorticity, yield interesting phenomena to fluid simulations.
The underlying discretization used in physical-based simulations ranges widely from Lagrangian particles and meshes to the Eulerian grid. By mapping a continuous fluid onto these structures, the fluid is tracked or observed by the discrete degrees of freedom in which those partial differential equations (PDEs) are solved to apply the physical force. To achieve better stability and accuracy, these constraints are typically solved implicitly, resulting in a large linear system to be solved using numerical solvers.
The contradiction between high resolution and high performance is a key issue in physical simulations. Typical approaches for improving performance include adaptivity[5-7], model reduction[8], and using accelerated solvers[9,10]. Other methods focus on small-scaled details, including turbulence synthesis[11], vortex sheets[12], turbulence particles[13], and multiscale solvers[14]. In these methods, solutions are proposed based on the white-box modeling of the problem; hence, their applicability might be limited.
Recently, emerging demands such as motion control, artistic stylization, and animation synthesis have resulted in high requirements for the simulation framework. In these applications, it is challenging to design intricate methods to satisfy the constraints or define the constraint explicitly.
Owing to rapid developments over the past decade, the data-driven method has become more effective and hence has been used widely in computer vision (CV) and natural language processing tasks. The data-driven method, particularly the machine learning (ML) method, relies on a large amount of historical data to build a mathematical model; it generates predictions based on the knowledge learned automatically from training. Treuille et al. first introduced the data-driven method into fluid simulation using a dimension reduction method to accelerate the complex fluid dynamic system[15]. Other data-driven models, including state graphs[16], regression forests[17], and neural networks[1821], have been applied to predict fluid evolution without numerical simulation. Based on CV concepts, data-driven methods provide new tools for studies associated with solver acceleration and detail enhancement, including autoencoders[19], long short-term memory (LSTM)[22], generative adversarial networks (GANs)[23], and continuous convolution[24]. Furthermore, they have prompted a broad array of applications of fluid simulation, such as fluid-style transfer[25,26], fluid reconstruction[27], game control[28], and turbulence modeling[29]. Compared with previous studies that summarize the use of data-driven methods in physical-based modeling[30,31], this survey primarily focuses on the integration of ML techniques in fluid simulation in computer graphics.
In this survey, we briefly introduce physical-based simulations for fluid animation in Section 2. Three sets of simulation results using traditional methods are shown in Figure 1. Section 3 describes the data-driven methodologies used in Eulerian, Lagrangian, and hybrid viewpoints as well as issues associated with spatial discretization. Several mainstream applications are reviewed in Section 4, including data-driven solvers, detail enhancement, animation synthesis, fluid control, and differentiable simulation. Typical technical issues in data-driven fluid simulations are discussed in Section 5.
2 Physical-based fluid simulation method
The incompressible fluid is governed by the well-known Navier-Stokes equation, which describes the fluid evolution in a continuous field during the simulation. Equation (1) shows the incompressible N-S equation.
D u D t = υ u - p ρ + f , u = 0
denotes the vector field of velocity,
the viscosity coefficient,
the density,
the scalar field of pressure, and
the body force. The first equation describes the momentum changes caused by viscosity, pressure, and body force. The second equation enforces the divergence-free condition on the velocity field to achieve incompressibility. A compressible variant can be derived based on the conservation of mass[35]. In computer graphics, the N-S equation is solved by discretizing the continuum from differential viewpoints, which will be discussed below.
2.1 Eulerian method
In the Eulerian method, fluid is observed at static points. It was first introduced into graphics by Foster et al.[36], and Stam[37] further improved its stability by introducing a semi-Lagrangian advection step. In Foster's and Stam's models, a fluid is discretized on a static unified grid. The physical properties are stored on the grid nodes and grid faces of a marker-and-cell (MAC) grid[38]. Subsequently, differential stencils are built on adjacent cells base on the center-differential scheme. The time integration in the Navier-Stokes equation is segregated and solved step by step. Specifically, the projection term is solved implicitly as a global Poisson equation of the pressure field, and it is typically the bottleneck of the entire simulation pipeline. A set of adaptivity strategies, including tall cells[39], far-field grids[40], Octree[5], and adaptive staggered-tilted grids[41], was proposed to capture more small-scale details in the region of interest while reducing the overall number of degrees of freedom.
2.2 Lagrangian method
The smoothed particle hydrodynamics (SPH) method was first proposed by Monaghan et al.[42] and applied to an interactive application by Müller et al.[43]. Macklin et al. further used the SPH kernel in position-based dynamics (PBD)[44], which is widely used in real-time multimaterial applications[45], to simulate a position-based fluid.
Lagrangian particles achieve mass conservation naturally while differential operators are harder to construct. Therefore, most of the differential stencils rely on the neighboring searching operation to approximate a local continuous field. In SPH, differentiation is performed on neighboring particles using a smooth kernel that weights particles around in its region. Cornelis et al.[46] and Bender et al.[33] further solved the instability problem and achieved incompressibility. Similar meshless differentiation methods can be seen in reproducing kernel particle methods[47] and the element-free Galerkin method[48].
2.3 Hybrid method
The hybrid method incorporates both the Eulerian and Lagrangian methods, in which a structured grid is used for differentiation and geometry tracking for thin feature preservation. The basic idea is to perform kinematic procedures on particles and dynamic steps on the static grid simultaneously. Meanwhile, a particle-grid interpolation is used to transfers the physical properties back and forth.
Harlow et al. first applied the particle-in-cell (PIC) method to fluid simulation, in which excessive numerical dissipation is introduced[49]. Based on the PIC method, Brackbill et al. proposed the fluid implicit particle (FLIP) method, which transfers the update of the velocity field instead of itself to attenuate the numerical dissipation[50]. Zhu further combined these two methods with a blending factor[51]. Jiang et al. achieved a grid-particle interpolation step using an affine descriptor[52], whereas Fu et al.[53] and Hu et al.[34] improved energy and vorticity conservation by augmenting each particle with a polynomial and moving least square descriptor during particle-grid interpolation.
Compared with the pure Eulerian method, the hybrid method reduces the numerical dissipation introduced by grid interpolation during the advection step and achieves mass conservation naturally with particles. Compared with the Lagrangian method, it solves the implicit constraints with a global linear system defined on a regular structure and improves the accuracy.
3 Data-driven methodology
In this section, we introduce data-driven methods based on different spatial discretizations and discuss the underlying connection between fluid simulation and the data-driven method.
3.1 Eulerian data-driven methods
We first limit our discussion to the regular grid used in existing data-driven methods. The Eulerian method primarily uses the central difference scheme to discretize the differential operators on adjacent cells. The local data-driven solver based on the descriptor of adjacent cells is inspired by the Eulerian method. For example, Yang et al. proposed a patch-based projection artificial neural network solver with a local descriptor on each cell containing the local pressure, cell type, and velocity divergence of neighboring cells. A similar concept was used in the detail synthesis[54]. Chu et al. trained a patch-based convolutional descriptor to locally synthesize high-resolution details from their space-time flow repositories[55]. The synthesis results are shown in Figure 2.
In addition, the global data-driven method regards the entire fluid field as the input. Owing to the structural similarity between the Eulerian field and image, and the receptive field similarity between the convolution kernel and central difference scheme, a model based on a convolution neural network (CNN) that is typically used by the CV community has been introduced to the data-driven Eulerian solver[18,56,57].
In terms of three-dimensional (3D) simulation, the space complexity of a global input increases cubically with the field length, necessitating feature extraction or input segmentation methods. In earlier studies[15,58], principal component analysis was used to extract features. However, owing to its poor adaptivity to nonlinear problems, solutions with data-driven methods are often pursued. For example, Ling et al. embedded Galilean invariance extracted from original data fields into Reynolds stress anisotropy predictions to improve Reynolds-averaged Navier-Stokes (RANS) turbulence models[59]. Xiong et al. proposed a learning-based framework to extract discrete Lagrangian vortex particles as features from a continuous Eulerian flow field[60]. Currently, some researchers[19,22,28,6163] combine the architectures of autoencoders in deep learning to obtain more effective feature representations.
3.2 Lagrangian data-driven methods
Unlike the regular grid, Lagrangian particles are unstructured, rendering network modeling challenging. Stanton et al. were the first to propose a precomputed state graph for real-time fluid simulation, in which each graph vertex represents particle positions and velocities in one frame, and one edge represents a short transition between states[16]. Meanwhile, one of the most important characteristics of the Lagrangian method is local differentiation based on neighboring particles. The regression forest method is based on the state graph. It constructs regression trees by applying particles distribution and the velocity of neighboring particles as node attributes. Using the state of neighboring particles as input, the future motion state of the central particle is predicted by traversing the trees[17]. Regarding the similarity between unordered Lagrangian particle sets and the point cloud model, ideas from point cloud processing have been adopted in the Lagrangian system in a few cross-disciplinary studies. The continuous convolution (CConv) method[64] proposes a point cloud based convolutional operation with a spherical filter. Ummenhofer et al. integrated the network into a fluid simulation by applying a convolution operator on both fluid and boundary particles in the neighborhood under the SPH framework[24]. Their water simulation results compared with those of divergence-free smoothed particle hydrodynamics (DFSPH)[65] are shown in Figure 3. In the graph-based network[66,67], in terms of frequent geometry changes in the fluid, researchers must dynamically build edges in the graph between neighboring particles and extract topological features using graph convolution. Schenck et al. proposed a novel model known as SPNets, in which a ConvSP layer is constructed to compute fluid-particle-particle pairwise interactions, and a ConvSDF layer to compute particle-static-object interactions[68].
3.3 Hybrid data-driven methods
In hybrid fluid simulation, data-driven optimization can be applied to either Lagrangian particles or the Eulerian grid.
For the data-driven solver, similar to the Eulerian method, neural network models are often used to replace the time-consuming procedures. In a previous study[69] based on the material point method, the internal acceleration calculation in particles was replaced with a neural network. For the detail synthesis, particle-based reseeding is a better option. Neural network models can be used to modify physical variables in regions of interest to enrich fluid details under low resolutions. For example, the machine learning FLIP (MLFLIP)[70] can model the splashing details in the FLIP framework as a probability distribution problem with two networks, the simulation results of which are shown in Figure 4. In this example, a classifier was trained to identify the splashing details of the fluid, and the splash particle velocity was sampled using a modifier that inferred a normal distribution for small-scale details.
4 Application
In this section, we present a wide range of applications of data-driven methods in fluid simulation, including data-driven solvers, detail enhancement, animation synthesis, fluid control, and differentiable simulation.
4.1 Data-driven solver
For the Eulerian method, some data-driven solvers have been proposed to replace the time-consuming projection procedure in the Eulerian method particularly to replace the numerical solver of the Poisson equation. Yang et al. proposed an artificial neural model for Poisson solving[54]. Subsequently, convolution-based projection solvers were used by Tompson et al.[18] and Xiao[56,57,71]. Others focused on the temporal continuity of the fluid field and modeled the time evolution of the field by introducing the recurrent neural network (RNN) structure and generative models. Wiewel et al. proposed an LSTM-based model with convolutional layers that can model projection solving with the pressure field as input, as well as model the entire simulation loop as the evolution of velocity fields[61]. Kim et al. encoded a physical simulation in their generative network model, which can generate plausible fluid results with divergence-free velocities under different parameters (Figure 5 (left))[62].
For Lagrangian particles, precomputed graphs[16,17] constructed with global or local particle distributions were proposed for motion prediction (Figure 5 (right)). To model the interparticle interaction force in the Lagrangian system, Battaglia et al. introduced an interaction network in a rigid-body simulation[72]. Li et al. developed dynamic particle interaction (DPI) networks that captured the dynamic, hierarchical, and long-range interactions of particles for fully learning particle dynamics[66]. This particle-based simulator can be applied to multimaterial scenarios. Sanchez-Gonzalez et al.[67] referred to graph networks (GNs)[73] and proposed the graph network-based simulator (GNS) to perform predictions in particle-based simulations. Compared with previous approaches, including DPI[66] and the GN-based model[73], the GNS method is simpler for simulation while affording better generalization. In addition, the continuous convolution method models the convolution operation on point clouds in a connection-free manner, and it can be regarded as an extension of the SPH operation. Ummenhofer et al. used spatial convolution to connect each liquid particle with its neighbors and obtained position correction by the continuous convolution of neighboring particles[24]. Mukherjee et al. used an LSTM-based RNN framework to replace the calculation of color field gradient at the droplet's contact front to address microscopic forces and boundary conditions when simulating small-scale fluids[74].
4.2 Detail enhancement
In contrast to the data-driven solver, detail enhancement adds fine detailed features to the physical field or visual result based on a coarse numerical simulation. It can be categorized into three major aspects: detail synthesis, super-resolution, and style transfer.
For a data-driven detail synthesis, a data repository of high-resolution details is typically used. During the simulation, those details are mapped to the corresponding patch on a low-resolution fluid for detail generation. Sato et al. synthesized small-scale details for 3D low-resolution fluid velocity fields in a patch-based manner by constructing a precomputed database of two-dimensional (2D) high-resolution fluid velocity fields and applying them in the advection step[58]. They improved their method by enriching low-resolution target smoke to one with turbulence details and using an optimization-based synthesis method to resolve the discontinuity between patch boundaries[75]. Chu et al. trained a CNN indicator to encode the similarity between fluid regions of different resolutions. Using this indicator, they successfully matched their simulation data to their reusable repositories of space-time flow data to optimize the visual results efficiently[55]. By contrast, Um et al. synthesized splashing droplets in the FLIP framework by first identifying the surface particles and predicting their velocities using a splashing distribution model[70].
As suggested by its name, "super-resolution" upscales the input flow field and recovers the high-resolution flow field by inferring the subgrid details in the original field. Xie et al. generated high-resolution fields from 3D volumetric smoke data using a GAN model with a temporal discriminator to maintain consistency in the details over time and a spatial discriminator to preserve the details of high spatial frequencies, whose super-resolution results are shown in Figure 6 [23]. Werhahn et al. improved scalability by decomposing the learning problem into multiple smaller subproblems[76].
In digital images and videos[7779], a style is transferred by generating a blending result containing the style of one input and the content of the other. Combined with fluid dynamics, style transfer creates interesting and impressive results. Kim et al. proposed a transport-based neural style transfer (TNST) model based on the Eulerian viewpoint[25]. They designed a differentiable renderer to stylize 3D smoke using 2D images ranging from simple patterns to intricate motifs. Subsequently, they extended their study to the Lagrangian representation by adding an additional transfer step between the particle and grid[26]. This method eliminated the expensive recursive alignment and ensured better time consistency than TNST.
4.3 Animation synthesis
In animation synthesis, the space-time evolution of a fluid is not solved explicitly but is predicted by the model based on the input. Interpolation-based methods generate animation with several specified animations by defining a smooth interpolation mapping between them. Thuerey used a keyframe-based method to interpolate a dense space-time deformation using an optical flow solver combined with a projection algorithm that recovered small-scale details[80]. Pan et al. used keyframe interpolation to generate fluid fields in between keyframes by obtaining the local driving force field of each frame through their space-time optimization method constrained by partial differential equations[81]. Sato et al. synthesized fluid animation by combining existing flow data in different scenarios. They defined an interpolation function to adjust the velocity at the boundary of different flow fields to obtain a natural combination with different flow fields[82].
For fluid reconstruction from video, Eckert et al. reconstructed a time-consistent flow field from a real-world smoke video, as shown in Figure 7, and proposed the estimation of the unseen flow region and an optimization scheme constrained by the real fluid[27]. They first initialized the density with a single pass of regular tomography and a velocity of zero; subsequently, they calculated the residual velocity that is necessary to match the motion and shape from the input image sequence, and the density inflow, while preserving the divergence-free constraint.
4.4 Fluid control
Fluid control primarily includes fluid editing, shape correction, and fluid motion control. For fluid editing, Flynn et al. applied the seam carving method from image processing to the post-processing of fluid animation, the results of which are shown in Figure 8 [83]. They defined a graph cut energy function based on the average curvature and kinetic energy; subsequently, the function was used to add or remove seams for the flow field to ensure the consistency of the content after the flow was resized. For shape correction, Nielsen et al. adopted the shape guidance method by adding constraints to guide a high-resolution fluid with a thin outer shell of liquid into a low-resolution fluid with a thicker shell of liquid to retain the low-resolution shape features in the high-resolution scene[84]. For fluid motion control, Ma et al. addressed fluid-solid coupling control in a data-driven manner[28]. They applied reinforcement learning to a game of jetting water onto a ball by inputting a combination of rigid body features and fluid features extracted using an autoencoder. Morton et al. transformed the original flow field coordinate into a new one using an encoder-decoder scheme to control vortex shedding suppression[85]. Holl et al. designed a control network that can be used to infer the control parameters to control fluid shape transformation[86].
4.5 Differentiable simulation
Differentiability has been widely investigated in recent years; it enables the derivative of a function to be evaluated numerically. By applying differentiability to a physical simulation, a simulator can be incorporated into the gradient-based optimization algorithm as well as combined with ML algorithms to effectively solve inverse problems. Schenck et al. proposed a differentiable fluid solver SPNets that can learn liquid parameters from data and perform liquid control tasks[68]. They added two new layers to the deep neural network (DNN) model: ConvSP and ConvSDF. The ConvSP layer was designed to compute the pressure correction solution in the PBF, whereas the ConvSDF layer was designed specifically to solve particle-static-object collisions. Hu et al. built a differentiable simulator for soft robotics to solve a series of inference, control, and collaboration tasks[87]. In addition, they integrated differentiability into the high-performance Taichi[88] framework (Figure 9), enabling differentiability for a rigid body, soft body, fluids, etc.[89]. In addition, Holl et al. designed a predictor network to plan the optimal trajectory and a control network to infer control parameters based on a differentiable simulator[86].
5 Technical issues and future directions
This section summarizes the typical technical issues from the various applications presented in Section 4. These issues necessitate future investigations.
5.1 Interpretability
The interpretability of the models, as a crucial problem in data-driven methods[29], has been discussed by Lipton[90]. Without interpretability, the data-driven method is a black-box model, whose validity is not guaranteed. He et al. investigated the relationship between a DNN and the linear finite element method to demonstrate the potential of the DNN in solving partial differential equations[91]. In their later publication, they discussed the close connection between the CNN and the multigrid method[92]. Yang et al. used a differentiable neural structure to learn a wide range of physical constraints, including rigid body rotations, rope, articulated body, collision, and contact[19].
5.2 Physical fidelity
Physical fidelity is achieved naturally by the law of physics in the physical-based method. By enforcing these constraints in the numerical algorithm, the fluid behaves the same as in the real world. However, in data-driven methods, fluid evolution is inferred based on historical data without physical constraints; this may result in deficient fidelity and visual artifacts.
To render ML models consistent with physical laws, physical knowledge has been incorporated into loss functions to enable models to capture generalizable dynamic patterns[31]. Notably, embedding the physical constraints, particularly the incompressibility, into the loss function is a typical approach in the Eulerian framework[56,57,71]. Kim et al. iteratively designed a loss function, which embedded incompressibility, better velocity extrapolation, and velocity gradients into their loss function[62].
Another approach for physics-based modeling is to design the network architecture based on physical models[31]. Owing to similarity in discretization, the CNN and continuous convolution network were used in the Eulerian and Lagrangian frameworks, respectively[24,56,57]. In addition, considering the spatial continuity of fluid simulation, the RNN is typically used for the prediction of consecutive frames[22,61,74].
In terms of generative models, Xie et al. introduced TempoGAN, which can generate high-resolution fields from 3D volumetric smoke data using the GAN with an additional discriminator network[23]. Werhahn et al. designed a temporal and spatial discriminator in a GAN to guarantee the time and space consistency of smoke to improve the physical fidelity in data-driven simulations[76]. More details regarding data-driven physics-based modeling are provided in[30,31].
5.3 PDE solving
As mentioned above, the data-driven solvers[18,54,56] demonstrate the potential ability of data-driven models for solving regular PDEs of the Poisson equation. Solving high-dimension PDEs with a low computation cost has been a longstanding problem. The data-driven method was leveraged in the numerical solver by Lagaris et al.[93]. Subsequent studies extend the data-driven PDE solver to higher-dimensional PDEs using deep learning techniques[94,95]. Another approach to address this issue is to enhance the numerical solver with a data-driven network; in this approach, convergence and correctness can be achieved easily using the base of the numerical solver[96]. By integrating an additional learned correction step into the differentiable pipeline to interact with the solver, Um et al. reduced numerical errors that were disregarded by the PDE solver[97]. This idea was adopted in the multigrid solver as well, where the network learns the optimal prolongation and restriction matrices[98,99]. In other studies, a network was proposed to learn the differential operator and uncover the underlying PDEs. For example, PDE-Net[100] learns the differential operators and the response function to uncover the underlying PDE.
5.4 Performance optimization
Parallel algorithms have been investigated in various fields of computer graphics[101103], including a CPU's multithreading and a GPU's parallel computing. They have been widely used in parallel training[104] and extended to physical calculations[105108] for acceleration.
In addition, Hu et al. designed an optimized compiler based on the characteristics of data structures and typical operations in a simulation to solve equations in a sparse computational domain[88]. Furthermore, they developed a differentiable programming language for simulation, thereby enabling task optimization, such as motion learning and motion control[89]. Additionally, such studies reveal a new research direction for optimizing the calculation at the operating system level, in which data-driven training processes can be accelerated or simulation processes regarded as data-driven optimization iterations.
5.5 Generalization
Generalization is an important indicator for evaluating data-driven models. Without generalization, the model tends to overfit, which is typically caused by limited data or complex network structures. In fluid simulation, generalization renders the model applicable to a broad range of fluid scenarios. Xiao et al. employed the incremental learning technique for a data-driven solver to improve generalization[56]. They evaluated the error between the network output and physical result for several frames. When the error exceeded a threshold, an online-learning loop was executed to fine-tune the trained model with the current scene.
5.6 Model reduction
Model reduction is a typical method for feature extraction in computer graphics and was first introduced to fluid simulation by Treuille et al.[15]. The main idea is to simplify the model for low computational complexity while preserving the features as much as possible. Wiewel et al. designed a CNN-based LSTM to extract fluid features in a latent space[61]. They further segmented their latent space into individual quantities, enabling them to alter the reductive quantities in the latent space without interfering with others[22].
5.7 Data processing
It is difficult to predict physical fields credibly by merely applying learning methods from raw fluid data[29]. Researchers often categorize the learning task into data processing, modeling, and evaluation. Among them, data processing is a basic yet significant task in data-driven fluid simulation methods for providing reasonable features to data-driven models. It includes fluid feature extraction, selection, identification, and measurement.
For feature extraction, spatial features are typically extracted from high-dimensional and complex fluid fields. For example, an autoencoder is used for encoding high-dimensional features into a latent space by training[22,28,61]. Xiong et al. extracted vortex features from a complex grid-based velocity field and represented vortices using learned Lagrangian particles, as discussed in Section 3.1[60].
For feature selection, Wang et al. adopted random forest regression to predict Reynolds stress discrepancies in RANS-based turbulence models, in which mean flow features, which comprised the curvature, gradient of pressure, ratio of turbulence production/dissipation, etc., were used as the input[109]. Milani et al. used the random forest model to predict the turbulent diffusivity field in film cooling flows[110]. They adopted the feature selection proposed by Ling[59] with an additional feature importance evaluation based on frequency.
For feature identification, Bao et al. preprocessed the input features of a model via feature identification, which includes feature defining, ranking, and optimization. They defined derivatives of variables and local physical parameters as features, ranked them using random forests, and optimized feature selection based on the normalized root mean square error and computational costs[111].
For feature measurement, the global distance error of data fields alone cannot be used to measure the similarity of local features. As for some structured data, such as grids, image similarity metrics can be borrowed from CV, and spatial operators such as convolution can be used to extract features for comparison. For some unstructured data, such as particles, neighborhood search operators can be used to extract spatial features for comparison. For example, Bao et al. designed a feature similarity measurement (FSM) to estimate the error in two-phase flows[111]. In addition, to overcome the difficulty of independently evaluating the mesh effect or model scalability in thermal-hydraulic simulations, Bao et al. proposed a data-driven approach based on FSM[111] to quantify the uncertainty simulation error by investigating the local similarity in multiscale data using ML[112]. Similarly, Kohl et al. proposed a stable and generalizing metric known as LSiM using a Siamese architecture to measure the similarities between two fluid fields or reality scenes[113].
5.8 Dataset framework
The dataset is the foundation for model training and evaluation. In CFD, Li et al. archived a dataset containing isotropic forced turbulence with the form of a time series of velocity and pressure fields, which has been widely used for turbulence modeling[114]. In addition, several CFD and fluid datasets including THT Lab (Myong et al.)[115], KTH Flow (Schlatter et al.)[116], and FDY DNS (Avsarkisov et al.)[117] only offer single simulated datasets with different resolutions. To provide numerous datasets for a single phenomenon, Eckert et al. constructed a comprehensive smoke plume dataset[27], which included the velocity field, density field, calibration data, rendered pictures, and videos, to satisfy the requirements of some reconstruction-from-video works[118,119]. Recently, a fully differentiable open-source PDE-solving toolkit known as PhiFlow 1 has been released. It can be used to execute a fully functional fluid simulation in a deep learning framework, such as TensorFlow[120] and Pytorch[121], on GPUs; furthermore, it has been adopted in previous studies[86,97].
6 Conclusion
Data-driven fluid simulation, which links ML and fluid animation, is highly promising. Such a simulation can produce detailed visual results in films, games, virtual reality, etc., without excessive computational overhead. Our survey focused on the methodologies and applications of data-driven methods in fluid animation. In addition, we discussed several issues and future directions pertaining to data-driven methods. Finally, we hope that this survey will provide basic understanding and inspiration for researchers interested in fluid animation.



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