The deep neural network solver for B-spline approximation

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.