Piecewise Polynomials (PPs) are utilized in several engineering disciplines,
like trajectory planning, to approximate position profiles given in the form of
a set of points. While the approximation target along with domain-specific
requirements, like Ck -continuity, can be formulated as a system of equations
and a result can be computed directly, such closed-form solutions posses
limited flexibility with respect to polynomial degrees, polynomial bases or
adding further domain-specific requirements. Sufficiently complex optimization
goals soon call for the use of numerical methods, like gradient descent. Since
gradient descent lies at the heart of training Artificial Neural Networks
(ANNs), modern Machine Learning (ML) frameworks like TensorFlow come with a set
of gradient-based optimizers potentially suitable for a wide range of
optimization problems beyond the training task for ANNs. Our approach is to
utilize the versatility of PP models and combine it with the potential of
modern ML optimizers for the use in function approximation in 1D trajectory
planning in the context of electronic cam design. We utilize available
optimizers of the ML framework TensorFlow directly, outside of the scope of
ANNs, to optimize model parameters of our PP model. In this paper, we show how
an orthogonal polynomial basis contributes to improving approximation and
continuity optimization performance. Utilizing Chebyshev polynomials of the
first kind, we develop a novel regularization approach enabling clearly
improved convergence behavior. We show that, using this regularization
approach, Chebyshev basis performs better than power basis for all relevant
optimizers in the combined approximation and continuity optimization setting
and demonstrate usability of the presented approach within the electronic cam
domain.

利用分段多项式来近似 1D 轨迹规划中电子凸轮设计的函数，结合现代机器学习优化器，特别是梯度下降方法，利用正交多项式基函数以及创新的正则化方法，在模型参数优化和连续性优化方面取得了明显提升的收敛性能。

机器学习优化的正交基分段多项式逼近

Machine Learning Optimized Orthogonal Basis Piecewise Polynomial  Approximation

We consider the task of estimating functions belonging to a specific class of
nonsmooth functions, namely so-called tame functions. These functions appear in
a wide range of applications: training deep learning, value functions of
mixed-integer programs, or wave functions of small molecules. We show that tame
functions are approximable by piecewise polynomials on any full-dimensional
cube. We then present the first ever mixed-integer programming formulation of
piecewise polynomial regression. Together, these can be used to estimate tame
functions. We demonstrate promising computational results.

我们考虑估计属于一类非光滑函数（称为 tame 函数）的函数的任务。我们展示了 tame 函数可以通过分段多项式在任何全维度立方体上进行逼近，并且提出了首个混合整数规划形式的分段多项式回归。这些方法可以用于估计 tame 函数，并展示了有希望的计算结果。