Ortho-ODE: Enhancing Robustness and of Neural ODEs against Adversarial Attacks

Neural Ordinary Differential Equations (NODEs) probed the usage of numerical solvers to solve the differential equation characterized by a Neural Network (NN), therefore initiating a new paradigm of deep learning models with infinite depth. NODEs were designed to tackle the irregular time series problem. However, NODEs have demonstrated robustness against various noises and adversarial attacks. This paper is about the natural robustness of NODEs and examines the cause behind such surprising behaviour. We show that by controlling the Lipschitz constant of the ODE dynamics the robustness can be significantly improved. We derive our approach from Grownwall's inequality. Further, we draw parallels between contractivity theory and Grownwall's inequality. Experimentally we corroborate the enhanced robustness on numerous datasets - MNIST, CIFAR-10, and CIFAR 100. We also present the impact of adaptive and non-adaptive solvers on the robustness of NODEs.