  Empirical adversarial risk minimization (EARM) is a widely used mathematical
framework to robustly train deep neural nets (DNNs) that are resistant to
adversarial attacks. However, both natural and robust accuracies, in
classifying clean and adversarial images, respectively, of the trained robust
models are far from satisfactory. In this work, we unify the theory of optimal
control of transport equations with the practice of training and testing of
ResNets. Based on this unified viewpoint, we propose a simple yet effective
ResNets ensemble algorithm to boost the accuracy of the robustly trained model
on both clean and adversarial images. The proposed algorithm consists of two
components: First, we modify the base ResNets by injecting a variance specified
Gaussian noise to the output of each residual mapping. Second, we average over
the production of multiple jointly trained modified ResNets to get the final
prediction. These two steps give an approximation to the Feynman-Kac formula
for representing the solution of a transport equation with viscosity, or a
convection-diffusion equation. For the CIFAR10 benchmark, this simple algorithm
leads to a robust model with a natural accuracy of {\bf 85.62}\% on clean
images and a robust accuracy of ${\bf 57.94 \%}$ under the 20 iterations of the
IFGSM attack, which outperforms the current state-of-the-art in defending
against IFGSM attack on the CIFAR10. Both natural and robust accuracies of the
proposed ResNets ensemble can be improved dynamically as the building block
ResNet advances. The code is available at:
\url{https://github.com/BaoWangMath/EnResNet}.
