We investigate finite-time Lyapunov exponents (FTLEs), a measure for exponential separation of input perturbations, of deep neural networks within the framework of continuous-depth neural ODEs. We demonstrate that FTLEs are powerful organizers for input-output dynamics, allowing for better interpretability and the comparison of distinct model architectures. We establish a direct connection between Lyapunov exponents and adversarial vulnerability, and propose a novel training algorithm that improves robustness by FTLE regularization. The key idea is to suppress exponents far from zero in the early stage of the input dynamics. This approach enhances robustness and reduces computational cost compared to full-interval regularization, as it avoids a full ``double'' backpropagation.

翻译：本文研究了连续深度神经ODE框架下深度神经网络的有限时间李雅普诺夫指数（FTLEs）——一种衡量输入扰动指数分离程度的指标。我们证明FTLEs是输入输出动力学的有效组织工具，能够提升模型可解释性并支持不同架构的比较。我们建立了李雅普诺夫指数与对抗脆弱性之间的直接关联，并提出一种通过FTLE正则化提升鲁棒性的新型训练算法。其核心思想是在输入动力学的早期阶段抑制远离零点的指数。相较于全区间正则化方法，该策略通过避免完整的“双重”反向传播过程，在提升鲁棒性的同时显著降低了计算成本。