Learning safe and stable robot motions from demonstrations remains a challenge, especially in complex, nonlinear tasks involving dynamic, obstacle-rich environments. In this paper, we propose Safe and Stable Neural Network Dynamical Systems S$^2$-NNDS, a learning-from-demonstration framework that simultaneously learns expressive neural dynamical systems alongside neural Lyapunov stability and barrier safety certificates. Unlike traditional approaches with restrictive polynomial parameterizations, S$^2$-NNDS leverages neural networks to capture complex robot motions, providing probabilistic guarantees through split conformal prediction in learned certificates. Experimental results in various 2D and 3D datasets -- including LASA handwriting and demonstrations recorded kinesthetically from the Franka Emika Panda robot -- validate the effectiveness of S$^2$-NNDS in learning robust, safe, and stable motions from potentially unsafe demonstrations. The source code, supplementary material and experiment videos can be accessed via https://github.com/allemmbinn/S2NNDS

翻译：从演示中学习安全稳定的机器人运动仍然是一个挑战，尤其是在涉及动态、障碍物丰富的复杂非线性任务中。本文提出安全稳定的神经网络动力学系统S$^2$-NNDS，这是一种从演示中学习的框架，能够同时学习表达能力强的神经动力学系统以及神经李雅普诺夫稳定性与障碍安全证书。与采用限制性多项式参数化的传统方法不同，S$^2$-NNDS利用神经网络捕捉复杂的机器人运动，并通过在学习到的证书中使用分割保形预测提供概率保证。在多种2D和3D数据集（包括LASA手写数据集以及通过Franka Emika Panda机器人运动示教记录的演示）上的实验结果验证了S$^2$-NNDS能够从潜在不安全的演示中学习到鲁棒、安全且稳定的运动。源代码、补充材料和实验视频可通过https://github.com/allemmbinn/S2NNDS访问。