In this work, we used deep neural networks (DNNs) to solve a fundamental problem in differential geometry. One can find many closed-form expressions for calculating curvature, length, and other geometric properties in the literature. As we know these concepts, we are highly motivated to reconstruct them by using deep neural networks. In this framework, our goal is to learn geometric properties from examples. The simplest geometric object is a curve. Therefore, this work focuses on learning the length of planar sampled curves created by a sine waves dataset. For this reason, the fundamental length axioms were reconstructed using a supervised learning approach. Following these axioms a simplified DNN model, we call ArcLengthNet, was established. The robustness to additive noise and discretization errors were tested.
翻译:在这项工作中,我们利用深神经网络来解决差异几何学中的一个根本问题。 人们可以在文献中找到许多用于计算曲线、长度和其他几何特性的封闭式表达式。 我们知道这些概念, 我们非常愿意通过使用深神经网络来重建这些概念。 在这个框架内, 我们的目标是从示例中学习几何特性。 最简单的几何对象是一个曲线。 因此, 这项工作侧重于学习由正弦波数据集生成的平面抽样曲线长度。 因此, 基本轴轴用一种受监督的学习方法进行了重建。 在这些轴后, 我们称之为ArcLengthNet 的简化 DNN 模型, 我们称之为ArcLength Net 。 测试了添加噪音和分解错误的坚固性。