Accelerating Eulerian Fluid Simulation With Convolutional Networks

Accelerating Eulerian Fluid Simulation With Convolutional Networks

1. 1. Introduction

   • Proposal
     • In this work we propose a data-driven solution to the invicid-Euler equations that is faster than traditional methods, while remaining competitive in longterm simulation stability and accuracy.
     • Use deep-learning techniques to derive an approximate solution the discrete Poisson equation
       • In this paper, we propose a machine learning based approach for accelerating this linear projection that is fast and whose complexity is data-independent. We leverage the power of deep-learning techniques to derive an approximate linear projection. We claim that our machine-learning approach can take advantage of the statistics of fluid data and the local sparsity structure of the discrete Poisson equation to learn an approximate methodth at ensures long-term stability using a fixed computational budget.
   • Main methods
     • Lagrangian
       • Approximate continuous quantities using discrete moving particles
     • Eulerian
       • Methods that approximate quantities on a fixed grid
       • The most demanding portion of this method is the “pressure projection” step
         • It involves solving the discrete Poisson equation
           • Exact solutions can be found via traditional convex optimization techniques, such as the Preconditioned Conjugate Gradient (PCG) algorithm or via stationary iterative methods, like the Jacobi or Gauss-Seidel methods. PCG exhibits fast asymptotic convergence, however it suffers from high time-constants and is generally notsuitedtoGPUhardware. The Jacobi method is used for real-time applications, how ever it suffers from poor asymptotic convergence.
   • Motivation
     • Real-time simulation of fluid flow is a long standing problem in many application domains: fromomputational fluid dynamics for industrial applications, to smoke and fluid effects for computer graphics and animation. High computational complexity of existing solutions has meant that real-time simulations have been possible under restricted conditions

2. 2. Related Work

   • A parallel multigrid poisson solver for fluids simulation on large grids. (McAdams et al., 2010)
     • Recent work has addressed the problem of computational performance in fluid simulations. (McAdams et al., 2010) proposed a multi-grid approach as a preprocessing step of the PCG method This method can significantly improve performance in large scene settings. However, multi-grid methods are very difficult to implement, hard to parallelize on the GPU, have non-trivial “failure-cases” (e.g. multigrid complexity resorts to single-resolution complexity for simulations with highly irregular domain boundaries), and the method still requires a iterative optimization procedure with data-dependent complexity. 
     • McAdams, Aleka, Sifakis, Eftychios, and Teran, Joseph. A parallel multigrid poisson solver for fluids simulation on large grids. In Proceedings of the 2010 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 65–74. Eurographics Association, 2010. 
     • multi-grid methods are very difficult to implement, hard to parallelize on the GPU
   • Low viscosity flow simulations for animation. (Molemaker et al., 2008)
     • Molemaker, Jeroen, Cohen, Jonathan M, Patel, Sanjit, and Noh, Jonyong. Low viscosity flow simulations for animation. In Proceedings of the 2008 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 9–18. Eurographics Association, 2008.
     • not exploit the statistics of the data to be processed
   • Model reduction for real-time fluids. (Treuille et al., 2006)
     • Treuille, Adrien, Lewis, Andrew, and Popovi´c, Zoran. Model reduction for real-time fluids. ACM Trans. Graph., 25(3):826–834, July 2006. ISSN 0730-0301. doi: 10.1145/1141911.1141962. URL http://doi. acm.org/10.1145/1141911.1141962. 
   • Blending liquids. (Raveendran et al., 2014)
   • Data-driven fluid simulations using regression forests. (Ladick´y et al.,l 2015)
     • Ladick´y et al., propose an adaptation of regression forests forsmoothedparticlehydrodynamics. Givenasetofhandcrafted features corresponding to individual forces, the authorstrainedaregressorforpredictingthestateofeachparticle in the following time-step
   • Datadriven projection method in fluid simulation. (Yang et al,, 2016)
     • Yang et al. train a patchbased neural network to predict the ground truth pressure given local previous-frame pressure, voxel occupancy, and velocity divergence of an input frame. 

3. 3. Fluid Equations

   • Fluid inviscid
     • Finite difference (FD) methods on a MAC grid (Harlow & Welch, 1965)
       • The MAC grid representation samples velocity components on the face of voxel cells, and the scalar quantities (e.g. pressure or density) at the voxel center.
     • Equations 1 and 2 can be solved via the standard operator splitting method, which involves an advection update and a “pressure projection” step [Algorithm 1]
     • For advection of scalar fields and self-advection of velocity, we perform a semi-Lagrangian backward particle trace using the Maccormack method
     • In our hybrid apporach we modify step 8 by replacing the exact projection for a learned one.
       • Rewriting the Poisson equation results in a large sparse linear system A*pt = b.
         • We use an alternative machine learning (and data-driven) approach to solving this linear system, where we train a ConvNet model to infer pt.

4. 4. Pressure Model

   • In traditional formulations, incompressibility is obtained only when the pressure is a solution of the linear system of equations given in Equation 3
     • In real-time applications, PCG or Jacobi iterations are truncated before reaching convergence. Therefore the obtained veloc- ity fields are divergence.
       • The fundamental observation is that, while there is no closed form solution and a numerical solution might be difficult to compute, the function mapping input data to the optimum of an optimization problem is deterministic. Therefore one can attempt to approximate it using a powerful regressor such as deep neural network
     • Instead of using supervised training to directly infer and score the next frame velocity (or pressure) - where the loss would be some distance measure to ground-truth data - we measure the squared L-2 norm of the divergence of the predicted velocity and minimize it directly (Equation 4).
       • The Equation 4 formulation has two significant benefits:
         • 1) since obtaining ground-truth velocity from expensive iterative methods is no longer necessary, we can perform non-trivial data-augmentation to the input velocity fields.
         • 2) we can incorporate loss information from a composition of multiple time-steps without the need of runningexact solvers for each frame.
           • We do so by stepping forward the simulation state from u0 for each training sample, to a small number of time-steps n in the future (i.e. we calculate uˆn).2

5. 5. Dataset Creationand Model Training

6. 6. Results and Analysis

7. 7.Conclusion