Boundary Graph Neural Networks for 3D Simulations

TL;DR

We address the challenge of representing complex particle-boundary interactions in 3D granular flows with triangular walls. We introduce Boundary Graph Neural Networks (BGNNs), which dynamically augment graphs with virtual boundary nodes and edges and compute shortest-distance interactions to enforce boundary conditions within a time-aware, scalable framework. An effective theory underpins the approach, including dynamic graph updates, distance-based messaging, and orientation-aware boundary features. Experiments on hoppers, rotating drums, and mixers demonstrate that BGNNs reproduce DEM trajectories within simulation uncertainties, generalize to out-of-distribution geometries and moving boundaries, and offer runtime advantages over traditional simulation.

Abstract

The abundance of data has given machine learning considerable momentum in natural sciences and engineering, though modeling of physical processes is often difficult. A particularly tough problem is the efficient representation of geometric boundaries. Triangularized geometric boundaries are well understood and ubiquitous in engineering applications. However, it is notoriously difficult to integrate them into machine learning approaches due to their heterogeneity with respect to size and orientation. In this work, we introduce an effective theory to model particle-boundary interactions, which leads to our new Boundary Graph Neural Networks (BGNNs) that dynamically modify graph structures to obey boundary conditions. The new BGNNs are tested on complex 3D granular flow processes of hoppers, rotating drums and mixers, which are all standard components of modern industrial machinery but still have complicated geometry. BGNNs are evaluated in terms of computational efficiency as well as prediction accuracy of particle flows and mixing entropies. BGNNs are able to accurately reproduce 3D granular flows within simulation uncertainties over hundreds of thousands of simulation timesteps. Most notably, in our experiments, particles stay within the geometric objects without using handcrafted conditions or restrictions.

Figures (16)

• Figure 1: Effective theories of gravitational planetary movement (left), and particle-boundary interactions (right). Planetary movement is fully described by Einstein's field equations that relate mass and energy densities to the curvature of spacetime. A much simpler but in most cases sufficient description is to apply Newton's law of gravity to representative point masses. Black arrows indicate progress in time. Analogously, the interactions of granular flow particles and boundary surface areas is modeled by an effective two-point interaction.
• Figure 2: Dynamic modification of the graph edges (red lines) and nodes (red points). Left: Calculation of the distances $\tilde{d}(v_0,\tilde{v}_0)$, $\tilde{d}(v_2,\tilde{v}_1)$ between real particle at nodes $v_0$, $v_2$ and the triangles corresponding to virtual particle nodes $\tilde{v}_0$, $\tilde{v}_1$. Right: Insertion of an additional edge between $\tilde{v}_0$ and $v_0$ and between $\tilde{v}_1$ and $v_2$ and representation of the nodes in terms of the corresponding node features $\mathbf{p}_{v_i}$, $\mathbf{x}_{v_i}$ and $\mathbf{\tilde{p}}_{\tilde{v}_j}$, $\mathbf{\tilde{x}}_{\tilde{v}_j}$ for real and virtual nodes.
• Figure 3: Hopper dynamics. Top: Distributions for cohesive and non-cohesive particles. Simulation data and BGNN predictions are compared. Particles are indicated by green spheres, triangular wall areas are yellow, the edges of these triangles are indicated by grey lines. In contrast to liquid-like non-cohesive particles, cohesive particles lead to congestion of the hopper. Bottom: Position (left) and flow profile (right) for non-cohesive particles. Corresponding plots for cohesive particles can be found in \ref{['sec:app_experiments']}. Simulation data (solid lines) and BGNN predictions (dashed lines) are compared. Simulation uncertainties are due to a change of the particle numbers ($\pm25\%$) and to different initial conditions. We provide simulation predictions for a hopper with more timesteps in animations at https://ml-jku.github.io/bgnn/.
• Figure 4: Rotating drum dynamics. Top: Particle distributions for cohesive and non-cohesive particles. Simulation data and BGNN predictions are compared. Particles are indicated by green spheres, triangular wall areas are yellow, the edges of these triangles are indicated by grey lines. The circular arrow indicates the rotation direction of the drum. In contrast to liquid-like non-cohesive particles, cohesive particles stick together much stronger. Bottom: Flow profile (left) and entropy plot (right) for non-cohesive particles. The entropy is shown for particle class assignment according to the x (blue) and z (red) position. Corresponding plots for cohesive particles can be found in \ref{['sec:app_experiments']}. Simulation data (solid lines) and BGNN predictions (dashed lines) are compared. Simulation uncertainties are due to a change of the particle numbers ($\pm25\%$) and to different initial conditions. We provide simulation predictions for a rotating drum with more timesteps in animations at https://ml-jku.github.io/bgnn/.
• Figure 5: OOD generalization behavior for the hopper (left) and the rotating drum (right). In contrast to the training and validation data the outlet size of the hopper was decreased, the inclination angles of the hopper side walls are enlarged, and, the length of the rotating drum is increased.