We introduce in this paper new and very effective numerical methods based on neural networks for the approximation of the mean curvature flow of either oriented or non-orientable surfaces. To learn the correct interface evolution law, our neural networks are trained on phase field representations of exact evolving interfaces. The structures of the networks draw inspiration from splitting schemes used for the discretization of the Allen-Cahn equation. But when the latter approximate the mean curvature motion of oriented interfaces only, the approach we propose extends very naturally to the non-orientable case. Through a variety of examples, we show that our networks, trained only on flows of smooth and simplistic interfaces, generalize very well to more complex interfaces, either oriented or non-orientable, and possibly with singularities. Furthermore, they can be coupled easily with additional constraints which opens the way to various applications illustrating the flexibility and effectiveness of our approach: mean curvature flows with volume constraint, multiphase mean curvature flows, numerical approximation of Steiner trees or minimal surfaces.
翻译:在本文中,我们引入了基于神经网络的新和非常有效的数字方法,以近似方向或非方向表面的平均曲线流。为了学习正确的界面进化法,我们的神经网络接受关于精确变化界面的相片外观的训练。网络结构从用于艾伦-卡恩方程式分解的分解计划中得到灵感。但是当后者接近方向界面的平均曲解运动时,我们建议的方法非常自然地延伸到非方向性的情况。我们通过多种例子显示我们的网络,仅受过关于光滑和简单化界面流的培训,非常接近更复杂的界面,要么面向方向,要么不面向方向,可能与奇特性。此外,它们很容易与额外的制约因素相结合,为显示我们方法的灵活性和有效性的各种应用打开了道路:具有体积制约、多阶段平均曲流、施泰纳树或最小表面的数值近似。