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Towards a graph neural network solver for granular dynamics

Martin Servin*†, Holger Götz†, Tomas Berglund□, and Erik Wallin†

† Department of Physics, Umeå University, SE-90187 Umeå, Sweden

* e-mail: martin.servin@umu.se, web page: http://digitalphysics.se

□ Algoryx Simulation AB, Umeå, Sweden. http://www.algoryx.se

ABSTRACT

The discrete element method (DEM) is a versatile but computationally intensive method for granular

dynamics simulation. We investigate the possibility of accelerating DEM simulations using graph

neural networks (GNN), which automatically support variable connectivity between particles. This

approach was recently found promising for particle-based simulation of complex fluids [1]. We start

from a time-implicit, or nonsmooth, DEM [2], where the computational bottleneck is the process of

solving a mixed linear complementarity problem (MLCP) to obtain the contact forces and particle

velocity update. This solve step is substituted by a GNN, trained to predict the MLCP solution.

Following [1], we employ an encoder-process-decoder structure for the GNN. The particle and

connectivity data is encoded in an input graph with particle mass, external force, and previous

velocity as node attributes, and contact overlap, normal, and tangent vectors as edge attributes. The

sought solution is represented in the output graph with the updated particle velocities as node

attributes and the contact forces as edge attributes. In the intermediate processing step, the input

graph is converted to a latent graph, which is then advanced with a fixed number of message passing

steps involving a multilayer perceptron neural network for updating the edge and node values. The

output graph, with the approximate solution to the MLCP, is finally computed by decoding the last

processed latent graph.

Both a supervised and unsupervised method are tested for training the network on granular

simulation of particles in a rotating or static drum. AGX Dynamics [3] is used for running the

simulations, and Pytorch [4] in combination with the Deep Graph Library [5] for the learning. The

supervised model learns from ground truth MLCP solutions, computed using a projected Gauss-

Seidel (PGS) solver, sampled from 1200 simulations involving 50-150 particles. The unsupervised

model learns to minimize a loss function derived from the MLCP residual function using particle

configurations extracted from the same simulations but ignoring the approximate solution from the

PGS solver. The simulation samples are split into training data (80%), validation data (10%), and test

data (10%). Network hyperparameter optimization is performed. The supervised GNN solver reaches

an error level of 1% for the contact forces and 0.01% on the particle velocities for a static drum. For a

rotating drum, the respective errors are 10% and 1%. The unsupervised GNN solver reaches 1%

velocity errors, 5% normal forces errors, but it has significant problems with predicting the friction

forces. The latter is presumably because of the discontinuous loss function that follows from the

Coulomb friction law and therefore we explore regularization of it. Finally, we discuss the potential

scalability and performance for large particle systems.

REFERENCES

[1] A. Sanchez-Gonzalez, J. Godwin, T. Pfaff, R. Ying, J. Leskovec, and P. Battaglia. “Learning to

Simulate Complex Physics with Graph Networks”, Int. Conf. M L. arXiv:2002.09405 (2020).

[2] M. Servin, D. Wang, C. Lacoursière, and K. Bodin. “Examining the smooth and nonsmooth

discrete element approaches to granular matter”. Int. J. Num. Meth. Engng., 97, 878-902 (2014).

[3] AGX Dynamics, https://www.algoryx.se/agx-dynamics, Accessed: 2020-03-13.

[4] A. Paszke et al. “PyTorch: An imperative style, high-performance deep learning library”,

NeurIPS, 8024-8035 (2019).

[5] M. Wang et al. “Deep graph library: A graph-centric, highly-performant package for graph

neural networks”, arXiv:1909.01315 (2019).