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：数据的丰富性在自然科学和工程领域为机器学习提供了巨大动力，然而物理过程的建模通常十分困难。其中一个尤为棘手的问题是如何高效表示几何边界。三角化几何边界在工程应用中已被充分理解且广泛使用。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一直极具挑战。在本工作中，我们引入了一种有效理论来建模粒子-边界相互作用，并由此提出了新型边界图神经网络（BGNNs），该网络能够动态修改图结构以符合边界条件。新型BGNNs在料斗、转鼓和混合机等复杂三维颗粒流过程中进行了测试，这些过程虽是现代工业机械的标准组件，却仍具有复杂几何结构。我们从计算效率以及对颗粒流和混合熵的预测精度两方面评估了BGNNs。BGNNs能够在数十万模拟时间步内，在模拟不确定性范围内精确再现三维颗粒流。尤为值得注意的是，在我们的实验中，粒子无需借助手工设定的条件或约束，便始终维持在几何对象内部。