Neural networks are vulnerable to adversarial input perturbations imperceptible to human, which calls for robust machine learning for safety-critical applications. In this paper, we propose a new neural ODE layer which is inspired by Hopfield-type neural networks. We prove that the proposed ODE layer has global asymptotic stability on the projected space, which implies the existence and uniqueness of its steady state. We further show that the proposed layer satisfies the local stability condition such that the output is Lipschitz continuous in the ODE layer input, guaranteeing that the norm of perturbation on the hidden state does not grow over time. By experiments we show that an appropriate level of stability constraints imposed on the proposed ODE layer can improve the adversarial robustness of ODE layers, and present a heuristic method for finding good hyperparameters for stability constraints.

神经网络容易受到人类无法察觉的对抗性输入扰动的影响，这需要对安全关键应用程序进行稳健的机器学习。在本文中，我们提出了一种受 Hopfield 型神经网络启发的新神经 ODE 层。我们证明了所提出的 ODE 层在投影空间上具有全局渐近稳定性，这意味着其稳态的存在性和唯一性。我们进一步表明，所提出的层满足局部稳定性条件，使得输出在 ODE 层输入中是 Lipschitz 连续的，保证隐藏状态的扰动范数不会随着时间的推移而增长。通过实验，我们表明对所提出的 ODE 层施加适当水平的稳定性约束可以提高 ODE 层的对抗性鲁棒性，