Yifei Duan

Yifei Duan, Li'ang Li, Guanghua Ji et al.

Deep learning has made significant applications in the field of data science and natural science. Some studies have linked deep neural networks to dynamic systems, but the network structure is restricted to the residual network. It is known that residual networks can be regarded as a numerical discretization of dynamic systems. In this paper, we back to the classical network structure and prove that the vanilla feedforward networks could also be a numerical discretization of dynamic systems, where the width of the network is equal to the dimension of the input and output. Our proof is based on the properties of the leaky-ReLU function and the numerical technique of splitting method to solve differential equations. Our results could provide a new perspective for understanding the approximation properties of feedforward neural networks.

Xiaoxuan Sun, Yifei Duan, Yingnan Deng et al.

In this paper, an operating system scheduling algorithm based on Double DQN (Double Deep Q network) is proposed, and its performance under different task types and system loads is verified by experiments. Compared with the traditional scheduling algorithm, the algorithm based on Double DQN can dynamically adjust the task priority and resource allocation strategy, thus improving the task completion efficiency, system throughput, and response speed. The experimental results show that the Double DQN algorithm has high scheduling performance under light load, medium load and heavy load scenarios, especially when dealing with I/O intensive tasks, and can effectively reduce task completion time and system response time. In addition, the algorithm also shows high optimization ability in resource utilization and can intelligently adjust resource allocation according to the system state, avoiding resource waste and excessive load. Future studies will further explore the application of the algorithm in more complex systems, especially scheduling optimization in cloud computing and large-scale distributed environments, combining factors such as network latency and energy efficiency to improve the overall performance and adaptability of the algorithm.

Yifei Duan, Yongqiang Cai

The universal approximation property (UAP) holds a fundamental position in deep learning, as it provides a theoretical foundation for the expressive power of neural networks. It is widely recognized that a composition of linear and nonlinear functions, such as the rectified linear unit (ReLU) activation function, can approximate continuous functions on compact domains. In this paper, we extend this efficacy to a scenario containing dynamical systems with controls. We prove that the control family $\mathcal{F}_1$ containing all affine maps and the nonlinear ReLU map is sufficient for generating flow maps that can approximate orientation-preserving (OP) diffeomorphisms on any compact domain. Since $\mathcal{F}_1$ contains only one nonlinear function and the UAP does not hold if we remove the nonlinear function, we call $\mathcal{F}_1$ a minimal control family for the UAP. On this basis, several mild sufficient conditions, such as affine invariance, are established for the control family and discussed. Our results reveal an underlying connection between the approximation power of neural networks and control systems and could provide theoretical guidance for examining the approximation power of flow-based models.

Li'ang Li, Yifei Duan, Guanghua Ji et al.

The study of universal approximation properties (UAP) for neural networks (NN) has a long history. When the network width is unlimited, only a single hidden layer is sufficient for UAP. In contrast, when the depth is unlimited, the width for UAP needs to be not less than the critical width $w^*_{\min}=\max(d_x,d_y)$, where $d_x$ and $d_y$ are the dimensions of the input and output, respectively. Recently, \cite{cai2022achieve} shows that a leaky-ReLU NN with this critical width can achieve UAP for $L^p$ functions on a compact domain ${K}$, \emph{i.e.,} the UAP for $L^p({K},\mathbb{R}^{d_y})$. This paper examines a uniform UAP for the function class $C({K},\mathbb{R}^{d_y})$ and gives the exact minimum width of the leaky-ReLU NN as $w_{\min}=\max(d_x,d_y)+Δ(d_x, d_y)$, where $Δ(d_x, d_y)$ is the additional dimensions for approximating continuous functions with diffeomorphisms via embedding. To obtain this result, we propose a novel lift-flow-discretization approach that shows that the uniform UAP has a deep connection with topological theory.