Jizu Huang

Chupeng Ma, Liqun Cao, Jizu Huang et al.

In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell--Schrödinger equations in the Coulomb gauge. We first prove the global existence of weak solutions to the equations. Next we propose an energy-conserving fully discrete finite element scheme for the system and prove the existence and uniqueness of solutions to the discrete system. The optimal error estimates for the numerical scheme without any time-step restrictions are then derived. Numerical results are provided to support our theoretical analysis.

Ying Wei, Chao Yang, Jizu Huang

In this paper, we present a parallel numerical algorithm for solving the phase field crystal equation. In the algorithm, a semi-implicit finite difference scheme is derived based on the discrete variational derivative method. Theoretical analysis is provided to show that the scheme is unconditionally energy stable and can achieve second-order accuracy in both space and time. An adaptive time step strategy is adopted such that the time step size can be flexibly controlled based on the dynamical evolution of the problem. At each time step, a nonlinear algebraic system is constructed from the discretization of the phase field crystal equation and solved by a domain decomposition based, parallel Newton--Krylov--Schwarz method with improved boundary conditions for subdomain problems. Numerical experiments with several two and three dimensional test cases show that the proposed algorithm is second-order accurate in both space and time, energy stable with large time steps, and highly scalable to over ten thousands processor cores on the Sunway TaihuLight supercomputer.

Kun Wang, Jizu Huang

In the paper, we present a high order fast algorithm with almost optimum memory for the Caputo fractional derivative, which can be expressed as a convolution of $u'(t)$ with the kernel $(t_n-t)^{-α}$. In the fast algorithm, the interval $[0,t_{n-1}]$ is split into nonuniform subintervals. The number of the subintervals is in the order of $\log n$ at the $n$-th time step. The fractional kernel function is approximated by a polynomial function of $K$-th degree with a uniform absolute error on each subinterval. We save $K+1$ integrals on each subinterval, which can be written as a convolution of $u'(t)$ with a polynomial base function. As compared with the direct method, the proposed fast algorithm reduces the storage requirement and computational cost from $O(n)$ to $O((K+1)\log n)$ at the $n$-th time step. We prove that the convergence rate of the fast algorithm is the same as the direct method even a high order direct method is considered. The convergence rate and efficiency of the fast algorithm are illustrated via several numerical examples.

Junyuan He, Zhonghao Sun, Jizu Huang

Tree tensor networks (TTNs) provide a compact and structured representation of high-dimensional data, making them valuable in various areas of computational mathematics and physics. In this paper, we present a rigorous mathematical framework for expressing high-order derivatives of functional TTNs, both with or without constraints. Our framework decomposes the total derivative of a given TTN into a summation of TTNs, each corresponding to the partial derivatives of the original TTN. Using this decomposition, we derive the Taylor expansion of vector-valued functions subject to ordinary differential equation constraints or algebraic constraints imposed by Runge--Kutta (RK) methods. As a concrete application, we employ this framework to construct order conditions for RK methods. Due to the intrinsic tensor properties of partial derivatives and the separable tensor structure in RK methods, the Taylor expansion of numerical solutions can be obtained in a manner analogous to that of exact solutions using tensor operators. This enables the order conditions of RK methods to be established by directly comparing the Taylor expansions of the exact and numerical solutions, eliminating the need for mathematical induction. For a given function $\vector{f}$, we derive sharper order conditions that go beyond the classical ones, enabling the identification of situations where a standard RK scheme of order $p$ achieves unexpectedly higher convergence order for the particular function. These results establish new connections between tensor network theory and classical numerical methods, potentially opening new avenues for both analytical exploration and practical computation.

Junyuan He, Jizu Huang

Constructing explicit Runge--Kutta (ERK) methods with as few stages as possible for a given order is a classical problem in numerical analysis. In this work, we introduce a $Q$/$D$-space framework of sufficient order conditions for ERK methods. This framework generalizes Butcher's classical simplifying assumptions by reformulating them in terms of simplified $Q$- and $D$-spaces defined through their residual vectors. It yields sufficient conditions which, together with $B(p)$, ensure order $p$. It also leads to a recursive construction procedure for ERK methods of arbitrary even order, in which the Butcher coefficients are obtained from two structured linear systems. For every even order $p\ge 4$, the construction produces ERK methods with stage number $s(p)=(p^2-2p+8)/4$. This stage count has the same leading term as that of the classical Gragg families, while improving the linear term. The free parameters retained by the construction further provide a systematic framework for designing methods with enhanced stability and short-time accuracy.

Jizu Huang, Rukang You, Tao Zhou

Multi-scale deep neural networks (MscaleDNNs) with downing-scaling mapping have demonstrated superiority over traditional DNNs in approximating target functions characterized by high frequency features. However, the performance of MscaleDNNs heavily depends on the parameters in the downing-scaling mapping, which limits their broader application. In this work, we establish a fitting error bound to explain why MscaleDNNs are advantageous for approximating high frequency functions. Building on this insight, we construct a hybrid feature embedding to enhance the accuracy and robustness of the downing-scaling mapping. To reduce the dependency of MscaleDNNs on parameters in the downing-scaling mapping, we propose frequency-adaptive MscaleDNNs, which adaptively adjust these parameters based on a posterior error estimate that captures the frequency information of the fitted functions. Numerical examples, including wave propagation and the propagation of a localized solution of the schr$\ddot{\text{o}}$dinger equation with a smooth potential near the semi-classical limit, are presented. These examples demonstrate that the frequency-adaptive MscaleDNNs improve accuracy by two to three orders of magnitude compared to standard MscaleDNNs.

Jizu Huang, Rukang You, Tao Zhou

Tensor neural networks (TNNs) have demonstrated their superiority in solving high-dimensional problems. However, similar to conventional neural networks, TNNs are also influenced by the Frequency Principle, which limits their ability to accurately capture high-frequency features of the solution. In this work, we analyze the training dynamics of TNNs by Fourier analysis and enhance their expressivity for high-dimensional multi-scale problems by incorporating random Fourier features. Leveraging the inherent tensor structure of TNNs, we further propose a novel approach to extract frequency features of high-dimensional functions by performing the Discrete Fourier Transform to one-dimensional component functions. This strategy effectively mitigates the curse of dimensionality. Building on this idea, we propose a frequency-adaptive TNNs algorithm, which significantly improves the ability of TNNs in solving complex multi-scale problems. Extensive numerical experiments are performed to validate the effectiveness and robustness of the proposed frequency-adaptive TNNs algorithm.