We consider the problem of learning stabilizable systems governed by nonlinear state equation $h_{t+1}=\phi(h_t,u_t;\theta)+w_t$. Here $\theta$ is the unknown system dynamics, $h_t $ is the state, $u_t$ is the input and $w_t$ is the additive noise vector. We study gradient based algorithms to learn the system dynamics $\theta$ from samples obtained from a single finite trajectory. If the system is run by a stabilizing input policy, we show that temporally-dependent samples can be approximated by i.i.d. samples via a truncation argument by using mixing-time arguments. We then develop new guarantees for the uniform convergence of the gradients of empirical loss. Unlike existing work, our bounds are noise sensitive which allows for learning ground-truth dynamics with high accuracy and small sample complexity. Together, our results facilitate efficient learning of the general nonlinear system under stabilizing policy. We specialize our guarantees to entry-wise nonlinear activations and verify our theory in various numerical experiments

研究稳定非线性系统动力学学习问题，使用基于梯度的算法从单个有限轨迹的样本中学习系统动态，特别地，针对 entry-wise 非线性激活函数列出保证，通过数值实验验证了理论公式的正确性。

非渐近性和准确的学习非线性动力系统