Learning Mesh-Based Simulation with Graph Networks

Mesh-based simulations are central to modeling complex physical systems in many disciplines across science and engineering. Mesh representations support powerful numerical integration methods and their resolution can be adapted to strike favorable trade-offs between accuracy and efficiency. However, high-dimensional scientific simulations are very expensive to run, and solvers and parameters must often be tuned individually to each system studied. Here we introduce MeshGraphNets, a framework for learning mesh-based simulations using graph neural networks. Our model can be trained to pass messages on a mesh graph and to adapt the mesh discretization during forward simulation. Our results show it can accurately predict the dynamics of a wide range of physical systems, including aerodynamics, structural mechanics, and cloth. The model's adaptivity supports learning resolution-independent dynamics and can scale to more complex state spaces at test time. Our method is also highly efficient, running 1-2 orders of magnitude faster than the simulation on which it is trained. Our approach broadens the range of problems on which neural network simulators can operate and promises to improve the efficiency of complex, scientific modeling tasks.

[1]  James F. O'Brien,et al.  Adaptive anisotropic remeshing for cloth simulation , 2012, ACM Trans. Graph..

[2]  Razvan Pascanu,et al.  Relational inductive biases, deep learning, and graph networks , 2018, ArXiv.

[3]  Max Welling,et al.  Semi-Supervised Classification with Graph Convolutional Networks , 2016, ICLR.

[4]  Xiangyu Hu,et al.  Liquid Splash Modeling with Neural Networks , 2017, Comput. Graph. Forum.

[5]  Emo Welzl,et al.  Smallest enclosing disks (balls and ellipsoids) , 1991, New Results and New Trends in Computer Science.

[6]  Nils Thuerey,et al.  Deep Learning Methods for Reynolds-Averaged Navier–Stokes Simulations of Airfoil Flows , 2018, AIAA Journal.

[7]  J. Zico Kolter,et al.  Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction , 2020, ICML.

[8]  Jure Leskovec,et al.  Learning to Simulate Complex Physics with Graph Networks , 2020, ICML.

[9]  S. M. Ali Eslami,et al.  PolyGen: An Autoregressive Generative Model of 3D Meshes , 2020, ICML.

[10]  Karthik Duraisamy,et al.  Prediction of aerodynamic flow fields using convolutional neural networks , 2019, Computational Mechanics.

[11]  Yao Zhang,et al.  Application of Convolutional Neural Network to Predict Airfoil Lift Coefficient , 2017, ArXiv.

[12]  Jiajun Wu,et al.  Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids , 2018, ICLR.

[13]  N. Ramakrishnan,et al.  An analysis of springback in sheet metal bending using finite element method (FEM) , 2007 .

[14]  J. Alonso,et al.  SU2: An Open-Source Suite for Multiphysics Simulation and Design , 2016 .

[15]  Mathieu Aubry,et al.  A Papier-Mache Approach to Learning 3D Surface Generation , 2018, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition.

[16]  M. S. Albergo,et al.  Flow-based generative models for Markov chain Monte Carlo in lattice field theory , 2019, Physical Review D.

[17]  Vladlen Koltun,et al.  Lagrangian Fluid Simulation with Continuous Convolutions , 2020, ICLR.

[18]  Derek Nowrouzezahrai,et al.  Subspace neural physics: fast data-driven interactive simulation , 2019, Symposium on Computer Animation.

[19]  Leslie Pack Kaelbling,et al.  Graph Element Networks: adaptive, structured computation and memory , 2019, ICML.

[20]  James F. O'Brien,et al.  Folding and crumpling adaptive sheets , 2013, ACM Trans. Graph..

[21]  German Capuano,et al.  Smart finite elements: A novel machine learning application , 2019, Computer Methods in Applied Mechanics and Engineering.

[22]  Nils Thürey,et al.  Latent Space Physics: Towards Learning the Temporal Evolution of Fluid Flow , 2018, Comput. Graph. Forum.

[23]  Raia Hadsell,et al.  Graph networks as learnable physics engines for inference and control , 2018, ICML.

[24]  Patrick Gallinari,et al.  Deep learning for physical processes: incorporating prior scientific knowledge , 2017, ICLR.

[25]  Pushmeet Kohli,et al.  Unveiling the predictive power of static structure in glassy systems , 2020 .

[26]  James F. O'Brien,et al.  Dynamic local remeshing for elastoplastic simulation , 2010, ACM Trans. Graph..

[27]  S. Marburg,et al.  Computational acoustics of noise propagation in fluids : finite and boudary element methods , 2008 .

[28]  James F. O'Brien,et al.  Adaptive tearing and cracking of thin sheets , 2014, ACM Trans. Graph..

[29]  Wei Li,et al.  Convolutional Neural Networks for Steady Flow Approximation , 2016, KDD.

[30]  Jingtian Tang,et al.  3D direct current resistivity modeling with unstructured mesh by adaptive finite-element method , 2010 .

[31]  R. Ramamurti,et al.  Simulation of Flow About Flapping Airfoils Using Finite Element Incompressible Flow Solver , 2001 .

[32]  Hamouine Abdelmadjid,et al.  A state-of-the-art review of the X-FEM for computational fracture mechanics , 2009 .

[33]  Ah Chung Tsoi,et al.  The Graph Neural Network Model , 2009, IEEE Transactions on Neural Networks.

[34]  David Pardo,et al.  A self-adaptive goal-oriented hp-finite element method with electromagnetic applications. Part II: Electrodynamics , 2007 .

[35]  Barbara Solenthaler,et al.  Data-driven fluid simulations using regression forests , 2015, ACM Trans. Graph..

[36]  A. Vacavant,et al.  Reconstructions of Noisy Digital Contours with Maximal Primitives Based on Multi-Scale/Irregular Geometric Representation and Generalized Linear Programming , 2017 .

[37]  Pierre Vandergheynst,et al.  Geometric Deep Learning: Going beyond Euclidean data , 2016, IEEE Signal Process. Mag..

[38]  Paul S. Heckbert,et al.  A Pliant Method for Anisotropic Mesh Generation , 1996 .

[39]  Karthik Kashinath,et al.  Towards Physics-informed Deep Learning for Turbulent Flow Prediction , 2020, KDD.

[40]  Donghyun You,et al.  Data-driven prediction of unsteady flow over a circular cylinder using deep learning , 2018, Journal of Fluid Mechanics.

[41]  Huamin Wang,et al.  NNWarp: Neural Network-Based Nonlinear Deformation , 2020, IEEE Transactions on Visualization and Computer Graphics.

[42]  Gurtej Kanwar,et al.  Equivariant flow-based sampling for lattice gauge theory , 2020, Physical review letters.

[43]  Sara Hooker,et al.  The hardware lottery , 2020, Commun. ACM.

[44]  Daniel Cohen-Or,et al.  MeshCNN: a network with an edge , 2019, ACM Trans. Graph..

[45]  Christoph Schwarzbach,et al.  Three-dimensional adaptive higher order finite element simulation for geo-electromagnetics—a marine CSEM example , 2011 .

[46]  F. V. Antunes,et al.  A review on 3D-FE adaptive remeshing techniques for crack growth modelling , 2015 .

[47]  Hermann A. Maurer,et al.  New Results and New Trends in Computer Science , 1991, Lecture Notes in Computer Science.