Adversaries Meet Partial Differential Equations  A First Look at Fluidic Adversarial Robustification Machine Learning models are vulnerable to adversarial examples, viz. these models misclassify data points that are slightly deviated from the correctly classified examples drawn from the same data distribution. These adversarial examples are a result of and reveal inherent blind spots in our training algorithms. In this article, we proposed to build a blind robustification system which can be applied on all input samples such that it targets to retrieve the original labels of adversarial examples without causing changes to the prediction of nonadversarial samples. We first concretely propose an energy functional designed to target adversarial perturbation. We then derive its gradient descent flow equation using the EulerLagrange method. Following this, we derive an efficient discretization scheme for the proposed partial differential equations. Finally, we perform a thorough analysis of the performance of the proposed system by testing it for a stateofart network and a universal benchmarking dataset.

Generalized Adversarial Training in Riemannian Space Adversarial examples, referred to as augmented data points generated by imperceptible perturbations of input samples, have recently drawn much attention. Wellcrafted adversarial examples may even mislead stateoftheart deep neural network (DNN) models to make wrong predictions easily. To alleviate this problem, many studies have focused on investigating how adversarial examples can be generated andor effectively handled. All existing works tackle this problem in the Euclidean space. In this paper, we extend the learning of adversarial examples to the more general Riemannian space over DNNs. The proposed work is important in that (1) it is a generalized learning methodology since Riemmanian space will be degraded to the Euclidean space in a special case; (2) it is the first work to tackle the adversarial example problem tractably through the perspective of Riemannian geometry; (3) from the perspective of geometry, our method leads to the steepest direction of the loss function, by considering the second order information of the loss function. We also provide a theoretical study showing that our proposed method can truly find the descent direction for the loss function, with a comparable computational time against traditional adversarial methods. Finally, the proposed framework demonstrates superior performance over traditional counterpart methods, using benchmark data including MNIST, CIFAR10 and SVHN.