Guanghua Ji

Xuelong Gu, Guanghua Ji, Qi Wang

We propose several linear, fully decoupled numerical schemes with first- and second-order temporal accuracy for a novel Q-tensor-based two-phase hydrodynamic model describing the coupling of active nematic liquid crystal solutions with isotropic solid substrates. The model is derived from the generalized Onsager principle and includes nontrivial terms that contribute zero to the total free-energy dissipation. We prove that the proposed decoupled linear schemes are thermodynamically consistent at the discrete level. In the passive limit, the SGE-BDF1 and SGE-PDG schemes are unconditionally energy stable, while the SGE-BDF2 scheme is energy stable with respect to a modified energy under a standard boundedness assumption and a sufficiently large stabilization parameter. We perform extensive numerical simulations to investigate how activity and other model parameters affect active nematic fluid-solid interactions. Finally, we analyze the physical mechanisms underlying the observed behaviors, providing deeper insight into the dynamics of soft confined active nematic fluids.

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.

Qun Qiu, Wei Si, Guanghua Ji et al.

In this paper, we develop a set of efficient methods to compute stationary states of the spherical Landau-Brazovskii (LB) model in a discretization-then-optimization way. First, we discretize the spherical LB energy functional into a finite-dimensional energy function by the spherical harmonic expansion. Then five optimization methods are developed to compute stationary states of the discretized energy function, including the accelerated adaptive Bregman proximal gradient, Nesterov, adaptive Nesterov, adaptive nonlinear conjugate gradient and adaptive gradient descent methods. To speed up the convergence, we propose a principal mode analysis (PMA) method to estimate good initial configurations and sphere radius. The PMA method also reveals the relationship between the optimal sphere radius and the dominant degree of spherical harmonics. Numerical experiments show that our approaches significantly reduce the number of iterations and the computational time

Wenshuai Hu, Guanghua Ji, Xiao Li

The Smectic-A (SmA) phase is modeled by a modified Landau-de Gennes (mLdG) model proposed by Xia et al. [Phys. Rev. Lett., 126 (2021), 177801], in which a tensor order parameter Q for the orientational order is coupled with a real scalar $u$ characterizing the positional order. In this paper, we propose and analyze a novel, highly efficient, and unconditionally energy-stable numerical scheme for this coupled system by combining the generalized scalar auxiliary variable-exponential integrator (GSAV-EI) approach with a relaxed correction strategy. In particular, we reformulate the exponential time differencing time discretization into an equivalent quasi-implicit backward Euler-type structure, a pivotal step that eliminates the restrictive CFL mesh-ratio conditions of the original GSAV-EI method and enables a rigorous fully discrete error analysis. Theoretically, we rigorously establish the unconditional energy stability with respect to a modified discrete energy and the uniform boundedness of the numerical solutions Q, along with optimal error estimates in both time and space. Comprehensive numerical experiments are presented to demonstrate the accuracy, efficiency, and structural preservation of the algorithm, as well as its capability in capturing complex topological defect dynamics.

Wenshuai Hu, Guanghua Ji, Xiao Li

The Smectic-A (SmA) phase is modeled by a modified Landau-de Gennes (mLdG) model proposed by Xia et al. [Phys. Rev. Lett., 126 (2021), 177801], in which a tensor order parameter $\mathbf{Q}$ for the orientational order is coupled with a real scalar $u$ characterizing the positional order. In this paper, we propose and analyze a novel, highly efficient, and unconditionally energy-stable numerical scheme for this coupled system by combining the generalized scalar auxiliary variable-exponential integrator (GSAV-EI) approach with a relaxed correction strategy. In particular, we reformulate the exponential time differencing time discretization into an equivalent quasi-implicit backward Euler-type structure, a pivotal step that eliminates the restrictive CFL mesh-ratio conditions of the original GSAV-EI method and enables a rigorous fully discrete error analysis. Theoretically, we rigorously establish the unconditional energy stability with respect to a modified discrete energy and the uniform boundedness of the numerical solutions $\mathbf{Q}$, along with optimal error estimates in both time and space. Comprehensive numerical experiments are presented to demonstrate the accuracy, efficiency, and structural preservation of the algorithm, as well as its capability in capturing complex topological defect dynamics.

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.