Abstract: 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.