The deep neural network solver for B-spline approximation

• Published in: Computer aided design 2024-04, Vol.169, p.103668, Article 103668
• Main Authors: Wen, Zepeng, Luo, Jiaqi, Kang, Hongmei
• Format: Article
• Language: English
• Subjects: B-spline approximation; Deep learning; Deep neural network solver; Knot placement
• ISSN: 0010-4485
• DOI: 10.1016/j.cad.2023.103668

This paper introduces a novel unsupervised deep learning approach to address the knot placement problem in the field of B-spline approximation, called deep neural network solvers (DNN-Solvers). Given discrete points, the DNN acts as a solver for calculating knot positions in the case of a fixed knot number. The input can be any initial knots and the output provides the desirable knots. The loss function is based on the approximation error. The DNN-Solver converts the lower-dimensional knot placement problem, characterized as a nonconvex nonlinear optimization problem, into a search for suitable network parameters within a high-dimensional space. Owing to the over-parameterization nature, DNN-Solvers are less likely to be trapped in local minima and robust against initial knots. Moreover, the unsupervised learning paradigm of DNN-Solvers liberates us from constructing high-quality synthetic datasets with labels. Numerical experiments demonstrate that DNN-Solvers are excellent in both approximation results and efficiency under the premise of an appropriate number of knots. •We propose deep neural network solvers to address the knot placement problem in B-spline approximation. Numerical experiments demonstrate that DNN-Solvers exhibit robustness against initial knots and efficiently compute desirable knots when knot number is provided.•The unsupervised learning paradigm of DNN-Solvers obviates the necessity of constructing high-quality labeled datasets and effectively removes dimensional constraints on discrete points and input knots.•DNN-Solvers offer a new perspective on the knot placement problem from the standpoint of deep learning. We provide an in-depth analysis of DNN-Solvers, covering aspects related to approximation property, local minima and frequency analysis.