Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks

Deep neural networks (DNNs) are well-known to be vulnerable to adversarial\nattacks, where malicious human-imperceptible perturbations are included in the\ninput to the deep network to fool it into making a wrong classification. Recent\nstudies have demonstrated that neural Ordinary Differential Equations (ODEs)\nare intrinsically more robust against adversarial attacks compared to vanilla\nDNNs. In this work, we propose a stable neural ODE with Lyapunov-stable\nequilibrium points for defending against adversarial attacks (SODEF). By\nensuring that the equilibrium points of the ODE solution used as part of SODEF\nis Lyapunov-stable, the ODE solution for an input with a small perturbation\nconverges to the same solution as the unperturbed input. We provide theoretical\nresults that give insights into the stability of SODEF as well as the choice of\nregularizers to ensure its stability. Our analysis suggests that our proposed\nregularizers force the extracted feature points to be within a neighborhood of\nthe Lyapunov-stable equilibrium points of the ODE. SODEF is compatible with\nmany defense methods and can be applied to any neural network's final regressor\nlayer to enhance its stability against adversarial attacks.\n