# Data-driven simulation in fluids animation: A survey

School of Software, Shanghai Jiao Tong University, Shanghai 200240, China

Abstract

Keywords： Fluid simulation ; Data driven ; Machine learning

Content

^{[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.

^{[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.

^{[15]}. Other data-driven models, including state graphs

^{[16]}, regression forests

^{[17]}, and neural networks

^{[18‒21]}, 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.

^{[35]}. In computer graphics, the N-S equation is solved by discretizing the continuum from differential viewpoints, which will be discussed below.

^{[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.

^{[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.

^{[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]}.

^{[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.

^{[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.

^{[18,56,57]}.

^{[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

^{[19,22,28,61‒63]}combine the architectures of autoencoders in deep learning to obtain more effective feature representations.

^{[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]}.

^{[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.

^{[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]}.

^{[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]}.

^{[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]}.

^{[23]}. Werhahn et al. improved scalability by decomposing the learning problem into multiple smaller subproblems

^{[76]}.

^{[77‒79]}, 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.

^{[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]}.

^{[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.

^{[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]}.

^{[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]}.

^{[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]}.

^{[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]}.

^{[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]}.

^{[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]}.

^{[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.

^{[101‒103]}, 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

^{[105‒108]}for acceleration.

^{[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.

^{[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.

^{[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]}.

^{[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.

^{[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]}.

^{[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.

^{[111]}.

^{[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]}.

^{[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]}.

Reference

Fei F, Zhang J, Li J, Liu Z H. A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows. Journal of Computational Physics, 2020, 400: 108972 DOI:10.1016/j.jcp.2019.108972

Zhang J, Tian P, Yao S, Fei F. Multiscale investigation of Kolmogorov flow: From microscopic molecular motions to macroscopic coherent structures. Physics of Fluids, 2019, 31(8): 082008 DOI:10.1063/1.5116206

Zhang J, John B, Pfeiffer M, Fei F, Wen D S. Particle-based hybrid and multiscale methods for nonequilibrium gas flows. Advances in Aerodynamics, 2019, 1(1): 1–15 DOI:10.1186/s42774-019-0014-7

Zhang J, Önskog T. Langevin equation elucidates the mechanism of the Rayleigh-Bénard instability by coupling molecular motions and macroscopic fluctuations. Physical Review. E, 2017, 96(4): 043104 DOI:10.1103/physreve.96.043104

Losasso F, Gibou F, Fedkiw R. Simulating water and smoke with an octree data structure. In: ACM SIGGRAPH 2004 Papers on-SIGGRAPH '04. Los Angeles, California, New York, ACM Press, 2004, 457–462 DOI:10.1145/1186562.1015745

Berger M J, Colella P. Local adaptive mesh refinement for shock hydrodynamics. Journal of Computational Physics, 1989, 82(1): 64–84 DOI:10.1016/0021-9991(89)90035-1

Yan H, Wang Z Y, He J, Chen X, Wang C B, Peng Q S. Real-time fluid simulation with adaptive SPH. Computer Animation and Virtual Worlds, 2009, 20(2/3): 417–426 DOI:10.1002/cav.300

Ferstl F, Ando R, Wojtan C, Westermann R, Thuerey N. Narrow band FLIP for liquid simulations. Computer Graphics Forum, 2016, 35(2): 225–232 DOI:10.1111/cgf.12825

Chu J Y, Zafar N B, Yang X B. A schur complement preconditioner for scalable parallel fluid simulation. ACM Transactions on Graphics, 2017, 36(4): 1 DOI:10.1145/3072959.3126843

Gao M, Wang X L, Wu K, Pradhana A, Sifakis E, Yuksel C, Jiang C. GPU optimization of material point methods. ACM Transactions on Graphics, 2019, 37(6): 1–12 DOI:10.1145/3272127.3275044

Kim T, Thürey N, James D, Gross M. Wavelet turbulence for fluid simulation. ACM Transactions on Graphics, 2008, 27(3): 1–6 DOI:10.1145/1360612.1360649

Pfaff T, Thuerey N, Gross M. Lagrangian vortex sheets for animating fluids. ACM Transactions on Graphics, 2012, 31(4): 1–8 DOI:10.1145/2185520.2185608

Pfaff T, Thuerey N, Cohen J, Tariq S, Gross M. Scalable fluid simulation using anisotropic turbulence particles. In: ACM SIGGRAPH Asia 2010 papers on- SIGGRAPH ASIA '10. Seoul, South Korea, New York, ACM Press, 2010, 1–8 DOI:10.1145/1882262.1866196

Thürey N, Wojtan C, Gross M, Turk G. A multiscale approach to mesh-based surface tension flows. ACM Transactions on Graphics (TOG), 2010, 29(4): 1–10 DOI:10.1145/1778765.1778785

Treuille A, Lewis A, Popović Z. Model reduction for real-time fluids. ACM Transactions on Graphics, 2006, 25(3): 826–834 DOI:10.1145/1141911.1141962

Stanton M, Humberston B, Kase B, O'Brien J F, Fatahalian K, Treuille A. Self-refining games using player analytics. ACM Transactions on Graphics, 2014, 33(4): 1–9 DOI:10.1145/2601097.2601196

Ladický L, Jeong S, Solenthaler B, Pollefeys M, Gross M. Data-driven fluid simulations using regression forests. ACM Transactions on Graphics, 2015, 34(6): 1–9 DOI:10.1145/2816795.2818129

Tompson J, Schlachter K, Sprechmann P, Perlin K. Accelerating Eulerian fluid simulation with convolutional networks. International Conference on Machine Learning. 2017, 3424‒3433

Yang S, He X, Zhu B. Learning Physical Constraints with Neural Projections. Advances in Neural Information Processing Systems, 2020, 33

Zhang J, Ma W J. Data-driven discovery of governing equations for fluid dynamics based on molecular simulation. Journal of Fluid Mechanics, 2020, 892: A5 DOI:10.1017/jfm.2020.184

Brunton S L, Noack B R, Koumoutsakos P. Machine learning for fluid mechanics. Annual Review of Fluid Mechanics, 2020, 52(1): 477–508 DOI:10.1146/annurev-fluid-010719-060214

Wiewel S, Kim B, Azevedo V C, Solenthaler B, Thuerey N. Latent space subdivision: stable and controllable time predictions for fluid flow. Computer Graphics Forum, 2020, 39(8): 15–25 DOI:10.1111/cgf.14097

Xie Y, Franz E, Chu M Y, Thuerey N. tempoGAN. ACM Transactions on Graphics, 2018, 37(4): 1–15 DOI:10.1145/3197517.3201304

Ummenhofer B, Prantl L, Thuerey N, Koltun V. Lagrangian fluid simulation with continuous convolutions. International Conference on Learning Representations, 2020 DOI:10.1146/annurev.fluid.35.101101.161209

Kim B, Azevedo V C, Gross M, Solenthaler B. Transport-based neural style transfer for smoke simulations. ACM Transactions on Graphics, 2019, 38(6): 188 DOI:10.1145/3355089.3356560

Kim B, Azevedo V C, Gross M, Solenthaler B. Lagrangian neural style transfer for fluids. ACM Transactions on Graphics, 2020, 39(4): 52 DOI:10.1145/3386569.3392473

Eckert M L, Um K, Thuerey N. ScalarFlow: a large-scale volumetric data set of real-world scalar transport flows for computer animation and machine learning. ACM Transactions on Graphics (TOG), 2019, 38(6): 1‒16 DOI:10.1145/3355089.3356545

Ma P C, Tian Y S, Pan Z R, Ren B, Manocha D. Fluid directed rigid body control using deep reinforcement learning. ACM Transactions on Graphics, 2018, 37(4): 1–11 DOI:10.1145/3197517.3201334

Duraisamy K, Iaccarino G, Xiao H. Turbulence modeling in the age of data. Annual Review of Fluid Mechanics, 2019, 51(1): 357–377 DOI:10.1146/annurev-fluid-010518-040547

Chang C W, Dinh N T. Classification of machine learning frameworks for data-driven thermal fluid models. International Journal of Thermal Sciences, 2019, 135: 559–579 DOI:10.1016/j.ijthermalsci.2018.09.002

Willard J, Jia X W, Xu S M, Steinbach M, Kumar V. Integrating physics-based modeling with machine learning: a survey. 2020

NewYork, ACMPress, 2001,15–22 DOI:10.1145/383259.383260

Bender J, Koschier D. Divergence-free SPH for incompressible and viscous fluids. IEEE Transactions on Visualization and Computer Graphics, 2017, 23(3): 1193–1206 DOI:10.1109/tvcg.2016.2578335

Hu Y M, Fang Y, Ge Z H, Qu Z Y, Zhu Y X, Pradhana A, Jiang C. A moving least squares material point method with displacement discontinuity and two-way rigid body coupling. ACM Transactions on Graphics, 2018, 37(4): 1‒14 DOI:10.1145/3197517.3201293

Von Mises R, Geiringer H, Ludford G S. Mathematical theory of compressible fluid flow. Courier Corporation, 2004 DOI:10.1017/S002211205923033X

Foster N, Metaxas D. Controlling fluid animation. In: Proceedings Computer Graphics International. Hasselt and Diepenbeek, Belgium, IEEE, 1997, 178–188 DOI:10.1109/cgi.1997.601299

York New, Press ACM, 1999, 121–128 DOI:10.1145/311535.311548

Harlow F H, Welch J E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids, 1965, 8(12): 2182 DOI:10.1063/1.1761178

Chentanez N, Müller M. Real-time Eulerian water simulation using a restricted tall cell grid. In: ACM SIGGRAPH 2011 papers on-SIGGRAPH '11. Vancouver, British Columbia, Canada, New York, ACM Press, 2011, 1–10

Zhu B, Lu W L, Cong M, Kim B, Fedkiw R. A new grid structure for domain extension. ACM Transactions on Graphics, 2013, 32(4): 1–12

Xiao Y W, Chan S, Wang S Q, Zhu B, Yang X B. An adaptive staggered-tilted grid for incompressible flow simulation. ACM Transactions on Graphics, 2020, 39(6): 1‒15

Monaghan J J. Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics, 1992, 30(1): 543–574

Müller M, Charypar D, Gross M H. Particle-based fluid simulation for interactive applications. Symposium on Computer Animation, 2003, 154‒159

Macklin M, Müller M. Position based fluids. ACM Transactions on Graphics, 2013, 32(4): 1–12

Macklin M, Müller M, Chentanez N, Kim T Y. Unified particle physics for real-time applications. ACM Transactions on Graphics, 2014, 33(4): 1–12

Cornelis J, Ihmsen M, Peer A, Teschner M. IISPH-FLIP for incompressible fluids. Computer Graphics Forum, 2014, 33(2): 255–262

Liu W K, Jun S, Zhang Y F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 1995, 20(8/9): 1081–1106

Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256

Harlow F H. The particle-in-cell method for numerical solution of problems in fluid dynamics. Office of Scientific and Technical Information (OSTI), 1962

Brackbill J U, Ruppel H M. FLIP: a method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions. Journal of Computational Physics, 1986, 65(2): 314–343

Zhu Y N, Bridson R. Animating sand as a fluid. ACM Transactions on Graphics, 2005, 24(3): 965–972

Jiang C, Schroeder C, Selle A, Teran J, Stomakhin A. The affine particle-in-cell method. ACM Transactions on Graphics, 2015, 34(4): 1–10

Fu C Y, Guo Q, Gast T, Jiang C, Teran J. A polynomial particle-in-cell method. ACM Transactions on Graphics, 2017, 36(6): 222

Yang C, Yang X B, Xiao X Y. Data-driven projection method in fluid simulation. Computer Animation and Virtual Worlds, 2016, 27(3/4): 415–424

Chu M Y, Thuerey N. Data-driven synthesis of smoke flows with CNN-based feature descriptors. ACM Transactions on Graphics, 2017, 36(4): 1–14

Xiao X Y, Yang C, Yang X B. Adaptive learning-based projection method for smoke simulation. Computer Animation and Virtual Worlds, 2018, 29(3/4): e1837

IEEE transactions on visualization and computer graphics. IEEE Transactions on Visualization and Computer Graphics, 2018, 24(4): i–ii

Sato S, Morita T, Dobashi Y, Yamamoto T. A data-driven approach for synthesizing high-resolution animation of fire. DigiPro '12: Proceedings of the Digital Production Symposium, 2012, 37–42

Ling J L, Kurzawski A, Templeton J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. Journal of Fluid Mechanics, 2016, 807: 155–166

Xiong S Y, He X Z, Tong Y J, Zhu B. Neural vortex method: from finite Lagrangian particles to infinite dimensional eulerian dynamics. 2020

Wiewel S, Becher M, Thuerey N. Latent space physics: towards learning the temporal evolution of fluid flow. Computer Graphics Forum, 2019, 38(2): 71–82

Kim B, Azevedo V C, Thuerey N, Kim T, Gross M, Solenthaler B. Deep fluids: a generative network for parameterized fluid simulations. Computer Graphics Forum, 2019, 38(2): 59–70

Thuerey N, Weißenow K, Prantl L, Hu X Y. Deep learning methods for Reynolds-averaged navier-stokes simulations of airfoil flows. AIAA Journal, 2019, 58(1): 25–36

Wang S L, Suo S, Ma W C, Pokrovsky A, Urtasun R. Deep parametric continuous convolutional neural networks. In: 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition. Salt Lake City, UT, USA, IEEE, 2018, 2589–2597

Bender J, Koschier D. Divergence-free smoothed particle hydrodynamics. In: Proceedings of the 14th ACM SIGGRAPH/Eurographics Symposium on Computer Animation. Los Angeles California, New York, NY, USA, ACM, 2015, 147–155

Li Y, Wu J, Tedrake R, Tenenbaum J B, Torralba A. Learning particle dynamics for manipulating rigid bodies, deformable objects, and fluids. 2018

Sanchez-Gonzalez A, Godwin J, Pfaff T, Ying R, Leskovec J, Battaglia P W. Learning to simulate complex physics with graph networks. 2020

Schenck C, Fox D. SPNets: Differentiable fluid dynamics for deep neural networks. Conference on Robot Learning. 2018, 317‒335

Dwikatama P A, Dharma D, Kistijantoro A I. Fluid simulation based on material point method with neural network. In: 2019 International Conference of Artificial Intelligence and Information Technology (ICAIIT). Yogyakarta, Indonesia, IEEE, 2019, 244–249

Um K, Hu X Y, Thuerey N. Liquid splash modeling with neural networks. Computer Graphics Forum, 2018, 37(8): 171–182

Xiao X Y, Wang H, Yang X B. A CNN-based flow correction method for fast preview. Computer Graphics Forum, 2019, 38(2): 431–440

Battaglia P, Pascanu R, Lai M, Rezende D J, Kavukcuoglu K. Interaction networks for learning about objects, relations and physics. Proceedings of the 30th International Conference on Neural Information Processing Systems. 2016, 4509‒4517

Sanchez-Gonzalez A, Heess N, Springenberg J T, Merel J, Riedmiller M A, Hadsell R, Battaglia P. Graph Networks as Learnable Physics Engines for Inference and Control. ICML, 2018

Mukherjee R, Li Q, Chen Z, Chu S, Wang H. Neuraldrop: DNN-based simulation of small-scale liquid flows on solids. 2018

Sato S, Dobashi Y, Kim T, Nishita T. Example-based turbulence style transfer. ACM Transactions on Graphics, 2018, 37(4): 1–9

Werhahn M, Xie Y, Chu M Y, Thuerey N. A multi-pass GAN for fluid flow super-resolution. Proceedings of the ACM on Computer Graphics and Interactive Techniques, 2019, 2(2): 1–21

Gatys L, Ecker A, Bethge M. A neural algorithm of artistic style. Journal of Vision, 2016, 16(12): 326

Gatys L A, Ecker A S, Bethge M. Image style transfer using convolutional neural networks. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Las Vegas, NV, USA, IEEE, 2016, 2414–2423

Ruder M, Dosovitskiy A, Brox T. Artistic style transfer for videos. In: Lecture Notes in Computer Science. Cham: Springer International Publishing, 2016, 26–36

Thuerey N. Interpolations of smoke and liquid simulations. ACM Transactions on Graphics, 2017, 36(4): 1

Pan Z R, Manocha D. Efficient solver for spacetime control of smoke. ACM Transactions on Graphics, 2017, 36(4): 1

Sato S, Dobashi Y, Nishita T. Editing fluid animation using flow interpolation. ACM Transactions on Graphics, 2018, 37(5): 1–12

Flynn S, Egbert P, Holladay S, Morse B. Fluid carving. ACM Transactions on Graphics, 2019, 38(6): 1–14

Nielsen M B, Bridson R. Guide shapes for high resolution naturalistic liquid simulation. In: ACM SIGGRAPH 2011 papers on-SIGGRAPH '11. Vancouver, British Columbia, Canada, New York, ACM Press, 2011, 1–8

Morton J, Jameson A, Kochenderfer M J, Kochenderfer M J. Deep dynamical modeling and control of unsteady fluid flows. Advances in Neural Information Processing Systems. 2018, 9258‒9268

Holl P, Thuerey N, Koltun V. Learning to Control PDEs with Differentiable Physics. International Conference on Learning Representations. 2019

Hu Y M, Liu J C, Spielberg A, Tenenbaum J B, Freeman W T, Wu J J, Rus D, Matusik W. ChainQueen: a real-time differentiable physical simulator for soft robotics. In: 2019 International Conference on Robotics and Automation (ICRA). Montreal, QC, Canada, IEEE, 2019, 6265–6271

Hu Y M, Li T M, Anderson L, Ragan-Kelley J, Taichi Durand F.. ACM Transactions on Graphics, 2019, 38(6): 1–16

Hu Y, Anderson L, Li T M, Sun Q, Carr N, Ragan-Kelley J, Durand F. DiffTaichi: Differentiable Programming for Physical Simulation. International Conference on Learning Representations. 2019

Lipton Z C. The mythos of model interpretability. Queue, 2018, 16(3): 31–57

Sci J H. Relu deep neural networks and linear finite elements. Journal of Computational Mathematics, 2020, 38(3): 502–527

He J C, Xu J C. MgNet: a unified framework of multigrid and convolutional neural network. Science China Mathematics, 2019, 62(7): 1331–1354

Lagaris I E, Likas A, Fotiadis D I. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 1998, 9(5): 987–1000

Han J, Jentzen A, E W. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences of the United States of America, 2018, 115(34): 8505–8510

Sirignano J, Spiliopoulos K. DGM: a deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 2018, 375: 1339–1364

Hsieh J T, Zhao S, Eismann S, Mirabella L, Ermon S. Learning neural PDE solvers with convergence guarantees. International Conference on Learning Representations. 2018

Um K, Brand R, Fei Y R, Holl P, Thuerey N. Solver-in-the-loop: Learning from differentiable physics to interact with iterative PDE-solvers. Advances in Neural Information Processing Systems, 2020, 33

Katrutsa A, Daulbaev T, Oseledets I. Deep multigrid: learning prolongation and restriction matrices. 2017

Greenfeld D, Galun M, Basri R, Yavneh I, Kimmel R. Learning to optimize multigrid PDE solvers. International Conference on Machine Learning. PMLR, 2019, 2415‒2423

Long Z C, Lu Y P, Dong B. PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. Journal of Computational Physics, 2019, 399: 108925

Dharma D, Jonathan C, Kistidjantoro A I, Manaf A. Material point method based fluid simulation on GPU using compute shader. In: 2017 International Conference on Advanced Informatics, Concepts, Theory, and Applications (ICAICTA). Denpasar, IEEE, 2017, 1–6

Moreland K, Angel E. The FFT on a GPU. Proceedings of the ACM SIGGRAPH/EUROGRAPHICS conference on Graphics hardware. 2003, 112‒119

Kobbelt L, Botsch M. A survey of point-based techniques in computer graphics. Computers & Graphics, 2004, 28(6): 801–814

Cui H G, Zhang H, Ganger G R, Gibbons P B, Xing E P. GeePS: scalable deep learning on distributed GPUs with a GPU-specialized parameter server. In: Proceedings of the Eleventh European Conference on Computer Systems. London United Kingdom, New York, NY, USA, ACM, 2016, 1–16

McAdams A, Sifakis E, Teran J. A Parallel Multigrid Poisson Solver for Fluids Simulation on Large Grids. Symposium on Computer Animation. 2010, 65‒73

Liu H X, Mitchell N, Aanjaneya M, Sifakis E. A scalable schur-complement fluids solver for heterogeneous compute platforms. ACM Transactions on Graphics, 2016, 35(6): 201

Jung H R, Kim S T, Noh J, Hong J M. A heterogeneous CPU-GPU parallel approach to a multigrid Poisson solver for incompressible fluid simulation. Computer Animation and Virtual Worlds, 2013, 24(3/4): 185–193

Lentine M, Zheng W, Fedkiw R. A novel algorithm for incompressible flow using only a coarse grid projection. ACM Transactions on Graphics (TOG), 2010, 29(4): 1‒9

Wang J X, Wu J L, Xiao H. A physics-informed machine learning approach of improving RANS predicted Reynolds stresses. In: 55th AIAA Aerospace Sciences Meeting. Grapevine, Texas, Reston, Virginia, AIAA, 2017

Milani P M, Ling J L, Eaton J K. Physical interpretation of machine learning models applied to film cooling flows. Journal of Turbomachinery, 2019, 141(1): 011004

Bao H, Feng J Y, Dinh N, Zhang H B. Computationally efficient CFD prediction of bubbly flow using physics-guided deep learning. International Journal of Multiphase Flow, 2020, 131: 103378

Bao H, Dinh N, Lin L Y, Youngblood R, Lane J, Zhang H B. Using deep learning to explore local physical similarity for global-scale bridging in thermal-hydraulic simulation. Annals of Nuclear Energy, 2020, 147: 107684

Kohl G, Um K, Thuerey N. Learning Similarity Metrics for Numerical Simulations. International Conference on Machine Learning. PMLR, 2020, 5349‒5360

Li Y, Perlman E, Wan M P, Yang Y K, Meneveau C, Burns R, Chen S Y, Szalay A, Eyink G. A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. Journal of Turbulence, 2008, 9: N31

Myong H K, Kasagi N. A new approach to the improvement of k-ε turbulence model for wall-bounded shear flows. JSME International Journal Ser 2, Fluids Engineering, Heat Transfer, Power, Combustion, Thermophysical Properties, 1990, 33(1): 63–72

Schlatter P, Örlü R. Assessment of direct numerical simulation data of turbulent boundary layers. Journal of Fluid Mechanics, 2010, 659: 116–126

Avsarkisov V, Hoyas S, Oberlack M, García-Galache J P. Turbulent plane Couette flow at moderately high Reynolds number. Journal of Fluid Mechanics, 2014, 751: R1

Zang G M, Idoughi R, Wang C L, Bennett A, Du J G, Skeen S, Roberts W L, Wonka P, Heidrich W. TomoFluid: reconstructing dynamic fluid from sparse view videos. In: 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). Seattle, WA, USA, IEEE, 2020, 1867–1876

Wang Y H, Idoughi R, Heidrich W. Stereo Event-based Particle Tracking Velocimetry for 3D Fluid Flow Reconstruction. European Conference on Computer Vision. Springer, Cham, 2020, 36–53

Abadi M, Barham P, Chen J, Chen Z, Davis A, Dean J, Devin M, Ghemawat S, Irving G, Isard M, Kudlur M, Levenberg J, Monga R, Moore S, Murray D G, Steiner B, Tucker P, Vasudevan V, Warden P, Wicke M, Yu Y, Zheng X. Tensorflow: A system for large-scale machine learning. 12th {USENIX} symposium on operating systems design and implementation ({OSDI} 16). 2016, 265‒283

Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L, Desmaison A, Kopf A, Yang E, DeVito Z, Raison M, Tejani A, Chilamkurthy S, Steiner B, Fang L, Bai J, Chintala S. Pytorch: An imperative style, high-performance deep learning library. Advances In Neural Information Processing Systems, 2019, 8026‒8037