基于偏微分方程的神经网络鲁棒性模型

近年来，深度神经网络（Deep Neural Network, DNN）在语音识别、图像识别等应用上取得一系列突破性进展，引起了新一轮的人工智能革命。深度神经网络也随之成为当前研究的热点。随着研究的深入，深度神经网络的一些不足也逐渐暴露出来，在对抗攻击下的不稳定性就是其中之一。这种不稳定性大大限制了深度神经网络在高可靠性、高安全性要求的场景中的应用，已经成为当前DNN研究中亟待解决的问题。在本项目中，我们创造性的将DNN与微分流形和偏微分方程联系在一起。通过这种联系，DNN的稳定性就转化为微分流形和偏微分方程的正则性和稳定性，使得我们可以利用微分流形和偏微分方程丰富的理论结构和计算方法来构造和研究具有内在稳定性的神经网络模型，从而发展出在攻击之下具有良好鲁棒性的数学模型及其相对应的高效算法。传统的微分流形和偏微分方程与新兴的深度学习网络的交叉必将对深度神经网络的研究产生极大的推动作用。

Recently, Deep Neural Networks (DNN) have achieved great breakthrough in many applications, leads a new revolution of Artificial Intelligence. On the other hand, recent research also reveal that DNN has many drawbacks. Instability under adversarial attacks is one of them. Small variation of the input may lead great change of the output. This instability becomes a bottleneck of DNN in many applications requiring high reliability. In this project, we bridge DNN with manifold and differential equations. Using this connection, we can use the theory of differential manifold and differential equation to study the stability of DNN. Then we can develop mathematical models of DNN with intrinsic stability to attacks. Manifold and differential equations provide powerful mathematical tools for the study of DNNs, and will promote the theoretical study of DNNs significantly.