var-nmODE: Model with L2-stability based on nmODE for defending against adversarial attacks

Deep neural networks (DNN) have demonstrated remarkable performance in various applications. However, their performance is significantly influenced by a wide range of perturbations, particularly adversarial perturbations, especially adversarial perturbations, which are difficult to recognize by the naked eye but cause the network to produce incorrect classifications. Some studies have shown that ordinary differential equation (ODE) networks are inherently more robust to adversarial perturbations than general deep networks. nmODE (Neural Memory Ordinary Differential Equation) is a recently proposed artificial neural network model, which has strong nonlinearity. Despite its potential, nmODE still faces challenges in adversarial defense. In this paper, we propose a variant model of neural memory ordinary differential equations (var-nmODE) to defend against adversarial attacks. Based on the theoretical foundation, var-nmODE has $L_2$ stable mapping, which corresponds to authentication defense against $L_2$ adversarial perturbations. Further, we conduct adversarial training on the proposed model and show that var-nmODE has better performance through experiments than nmODE. In addition, through adversarial training, the performance of var-nmODE is significantly improved, which indicates that our proposed model can resist adversarial disturbance. It is worth mentioning that var-nmODE provides inherent and certified stability, making it a valuable addition to deep learning defense research.