We present a physics-informed machine learning (PIML) scheme for the feedback linearization of nonlinear discrete-time dynamical systems. The PIML finds the nonlinear transformation law, thus ensuring stability via pole placement, in one step. In order to facilitate convergence in the presence of steep gradients in the nonlinear transformation law, we address a greedy-wise training procedure. We assess the performance of the proposed PIML approach via a benchmark nonlinear discrete map for which the feedback linearization transformation law can be derived analytically; the example is characterized by steep gradients, due to the presence of singularities, in the domain of interest. We show that the proposed PIML outperforms, in terms of numerical approximation accuracy, the traditional numerical implementation, which involves the construction--and the solution in terms of the coefficients of a power-series expansion--of a system of homological equations as well as the implementation of the PIML in the entire domain, thus highlighting the importance of continuation techniques in the training procedure of PIML.

翻译：我们提出了一种基于物理信息机器学习（PIML）的方案，用于非线性离散时间动力系统的反馈线性化。该方案通过一步操作找到非线性变换律，从而通过极点配置确保系统稳定性。为应对非线性变换律中存在陡峭梯度时的收敛问题，我们提出了一种贪婪式训练流程。通过一个基准非线性离散映射（其反馈线性化变换律可解析推导）评估了所提PIML方法的性能；该示例在相关定义域内因存在奇点而具有陡峭梯度。我们证明：相比传统数值方法（需构建并求解幂级数展开系数的同调方程组）以及全域PIML实现，所提PIML在数值逼近精度上更优，揭示了连续技术（continuation techniques）在PIML训练流程中的重要性。