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:
https://github.com/BaoWangMath/EnResNet.

提出了一种基于传输方程最优控制理论的 ResNets 集合算法，该算法通过给每个残差映射的输出添加方差指定的高斯噪声的方式修改基本 ResNets 算法，并通过平均多个经过联合训练的修改后的 ResNets 产生的预测结果来提高抵抗 IFGSM 攻击的模型的预测准确率，该算法在 CIFAR10 基准测试中表现出相对于当前领先的防御 IFGSM 攻击的模型而言更高的训练鲁棒性以及自然准确性。

利用 Feynman-Kac 形式的 ResNets 集成来提高自然和鲁棒的准确性

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

In this article, we generalize the Wasserstein distance to measures with
different masses. We study the properties of such distance. In particular, we
show that it metrizes weak convergence for tight sequences.
We use this generalized Wasserstein distance to study a transport equation
with source, in which both the vector field and the source depend on the
measure itself. We prove existence and uniqueness of the solution to the Cauchy
problem when the vector field and the source are Lipschitzian with respect to
the generalized Wasserstein distance.

用广义的 Wasserstein 距离研究了源项的输运方程，证明了解的唯一性和存在性。