  Automatically determining knot number and positions is a fundamental and
challenging problem in B-spline approximation. In this paper, the knot
placement is abstracted as a mapping from initial knots to the optimal knots.
We innovatively introduce a deep neural network solver to approximate the
mapping. The neural network is composed of several subnetworks. Each subnetwork
is designed to approximate the optimal knot positions in the case of fixed knot
number. All the subnetworks are stacked together to find the optimal knots
(including knot number and knot positions) within some given tolerance. Owing
to the powerful approximation capabilities, as well as mature algorithms
developed in deep learning, the proposed method can effectively and efficiently
find the optimal knot number and knot positions. Several numerical examples are
demonstrated to show the superiority of our method.
