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.

翻译：数据的丰富性为自然科学和工程领域的机器学习注入了强大动力，但物理过程的建模往往仍面临困难。其中，几何边界的有效表征是一个尤为棘手的问题。三角化几何边界在工程应用中广为人知且普遍存在，然而，由于其在尺寸和方向上的异质性，将其整合到机器学习方法中历来十分困难。在本工作中，我们引入了一种有效的理论来建模粒子与边界间的相互作用，进而提出了新型边界图神经网络（BGNN），该网络能够动态调整图结构以满足边界条件。新型BGNN在料斗、滚筒及混合机等复杂三维颗粒流过程中进行了测试，这些过程虽为现代工业机械的标准组件，却仍具有复杂的几何结构。我们从计算效率以及颗粒流与混合熵的预测精度两方面对BGNN进行了评估。结果表明，BGNN能够在数十万模拟时间步长内，以模拟不确定性为基准，精确重现三维颗粒流。尤为显著的是，在我们的实验中，粒子无需借助人工设定条件或约束，即可始终保持在几何对象内部。