Boundary Graph Neural Networks for 3D Simulations

The abundance of data has given machine learning considerable momentum in natural sciences and engineering. However, the modeling of simulated physical processes remains difficult. A key problem is the correct handling of geometric boundaries. While triangularized geometric boundaries are very common in engineering applications, they are notoriously difficult to model by machine learning approaches due to their heterogeneity with respect to size and orientation. In this work, we introduce Boundary Graph Neural Networks (BGNNs), which dynamically modify graph structures to address boundary conditions. Boundary graph structures are constructed via modifying edges, augmenting node features, and dynamically inserting virtual nodes. The new BGNNs are tested on complex 3D granular flow processes of hoppers and rotating drums which are standard components of industrial machinery. Using precise simulations that are obtained by an expensive and complex discrete element method, BGNNs are evaluated in terms of computational efficiency as well as prediction accuracy of particle flows and mixing entropies. Even if complex boundaries are present, BGNNs are able to accurately reproduce 3D granular flows within simulation uncertainties over hundreds of thousands of simulation timesteps, and most notably particles completely stay within the geometric objects without using handcrafted conditions or restrictions.

[1]  William Kahan,et al.  Pracniques: further remarks on reducing truncation errors , 1965, CACM.

[2]  L. Fan,et al.  Application of a Discrete Mixing Model to the Study of Mixing of Multicomponent Solid Particles , 1975 .

[3]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[4]  L. Lucy A numerical approach to the testing of the fission hypothesis. , 1977 .

[5]  P. Cundall,et al.  A discrete numerical model for granular assemblies , 1979 .

[6]  Steve Plimpton,et al.  Fast parallel algorithms for short-range molecular dynamics , 1993 .

[7]  D. Sulsky Erratum: Application of a particle-in-cell method to solid mechanics , 1995 .

[8]  Andreas Klein,et al.  A Generalized Kahan-Babuška-Summation-Algorithm , 2005, Computing.

[9]  David Eberly,et al.  Distance Between Point and Triangle in 3D , 2008 .

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

[11]  Geoffrey E. Hinton,et al.  Rectified Linear Units Improve Restricted Boltzmann Machines , 2010, ICML.

[12]  M. V. D. Panne,et al.  Displacement Interpolation Using Lagrangian Mass Transport , 2011 .

[13]  T. Metzger,et al.  Moisture content and residence time distributions in mixed-flow grain dryers , 2011 .

[14]  C. Kloss,et al.  Models, algorithms and validation for opensource DEM and CFD-DEM , 2012 .

[15]  Geoffrey E. Hinton,et al.  ImageNet classification with deep convolutional neural networks , 2012, Commun. ACM.

[16]  A. Mangeney,et al.  Exact solution for granular flows , 2013 .

[17]  Miles Macklin,et al.  Position based fluids , 2013, ACM Trans. Graph..

[18]  Christophe Kassiotis,et al.  Unified semi-analytical wall boundary conditions in SPH: analytical extension to 3-D , 2014, Numerical Algorithms.

[19]  G. Lodewijks,et al.  DEM speedup: Stiffness effects on behavior of bulk material , 2014 .

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

[21]  Jian Sun,et al.  Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification , 2015, 2015 IEEE International Conference on Computer Vision (ICCV).

[22]  Jian Sun,et al.  Deep Residual Learning for Image Recognition , 2015, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[23]  Xavier Bresson,et al.  Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering , 2016, NIPS.

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

[25]  Andreas A. Aigner,et al.  fastDEM: A method for faster DEM simulations of granular media , 2017 .

[26]  Samuel S. Schoenholz,et al.  Neural Message Passing for Quantum Chemistry , 2017, ICML.

[27]  Andre Pradhana,et al.  A moving least squares material point method with displacement discontinuity and two-way rigid body coupling , 2018, ACM Trans. Graph..

[28]  R. Zemel,et al.  Neural Relational Inference for Interacting Systems , 2018, ICML.

[29]  T. Roessler,et al.  DEM parameter calibration of cohesive bulk materials using a simple angle of repose test , 2019, Particuology.

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

[31]  C. Coetzee Calibration of the discrete element method: Strategies for spherical and non-spherical particles , 2020 .

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

[33]  Max Welling,et al.  E(n) Equivariant Graph Neural Networks , 2021, ICML.

[34]  Nathanael Perraudin,et al.  Scalable Graph Networks for Particle Simulations , 2020, AAAI.

[35]  T. Pfaff,et al.  Learning Mesh-Based Simulation with Graph Networks , 2020, International Conference on Learning Representations.

[36]  Sepp Hochreiter,et al.  Learning 3D Granular Flow Simulations , 2021, ArXiv.