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Empirical adversarial risk minimization (EARM) is a widely used mathematicalframework to robustly train deep neural nets (DNNs) that are resistant toadversarial attacks. However, both natural and robust accuracies, inclassifying clean and adversarial images, respectively, of the trained robustmodels are far from satisfactory. In this work, we unify the theory of optimalcontrol of transport equations with the practice of training and testing ofResNets. Based on this unified viewpoint, we propose a simple yet effectiveResNets ensemble algorithm to boost the accuracy of the robustly trained modelon both clean and adversarial images. The proposed algorithm consists of twocomponents: First, we modify the base ResNets by injecting a variance specifiedGaussian noise to the output of each residual mapping. Second, we average overthe production of multiple jointly trained modified ResNets to get the finalprediction. These two steps give an approximation to the Feynman-Kac formulafor representing the solution of a transport equation with viscosity, or aconvection-diffusion equation. For the CIFAR10 benchmark, this simple algorithmleads to a robust model with a natural accuracy of {\bf 85.62}\% on cleanimages and a robust accuracy of ${\bf 57.94 \%}$ under the 20 iterations of theIFGSM attack, which outperforms the current state-of-the-art in defendingagainst IFGSM attack on the CIFAR10. Both natural and robust accuracies of theproposed ResNets ensemble can be improved dynamically as the building blockResNet advances. The code is available at:\url{https://github.com/BaoWangMath/EnResNet}.

ResNets Ensemble via the Feynman-Kac Formalism to Improve Natural and Robust Accuracies