Length Learning for Planar Euclidean Curves
This work addresses the fundamental problem of learning geometric properties, specifically curve length, for researchers in differential geometry and machine learning, representing an incremental step in applying DNNs to known mathematical concepts.
This paper explores the use of deep neural networks (DNNs) to learn the length of planar sampled curves, specifically those generated by sine waves. The authors reconstructed fundamental length axioms using a supervised learning approach and developed a simplified DNN model called ArcLengthNet.
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.