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劲动力，然而物理过程的建模往往困难重重。其中最具挑战性的问题之一是如何有效表示几何边界。三角化的几何边界在工程应用中广为人知且普遍存在，但由于其在尺寸和方向上的异质性，将其集成到机器学习方法中历来极为困难。本研究提出了一种有效理论来建模粒子-边界相互作用，并由此开发出新型边界图神经网络（BGNNs），该网络通过动态修改图结构来满足边界条件。新型BGNNs在料斗、转筒和混合器（均为现代工业机械的标准部件，但具有复杂几何形状）的复杂三维颗粒流过程中进行了测试。我们从计算效率以及颗粒流和混合熵的预测精度两方面评估了BGNNs的性能。实验表明，BGNNs能够在数十万模拟时间步长内准确复现三维颗粒流，且结果处于模拟不确定范围内。最值得注意的是，在我们的实验中，颗粒始终保持在几何对象内，无需借助人工设定的条件或限制。