arXiv subjects

Zeng Liu

Publications and source records attributed to Zeng Liu.

16 recordsLinked to original sources

arXiv2026

A Hybrid POD-Autoencoder Framework for Reduced Order Modeling of Turbulent Flow via Strategic Field Decomposition

This study proposes a hybrid reduced-order modeling (ROM) framework for the simulation of turbulent flow. The central idea is to decompose flow dynamics according to their temporal characteristics and predict the resulting components individually. The full field is first divided into a sub-field represented by a limited number of proper orthogonal decomposition (POD) modes (named as POD-retained field) and the corresponding residual sub-field (named as POD-truncated field). A frequency-informed POD strategy identifies the retained modes by considering both modal energy and dominant frequency. The evolution of retained POD coefficients, which feature similar temporal scales, is described using a vector autoregressive (VAR) model. In parallel, the POD-truncated field is compressed into a low-dimensional latent space using a Fourier-neural-operator-based Koopman $\beta$-variational autoencoder (FK-$\beta$-VAE), with the latent variables subsequently predicted by a switching-VAR model. Turbulent statistics of the full field are recovered by combining the contributions from the two components. The framework is assessed using turbulent channel flow at a friction Reynolds number of $110$. The predicted Reynolds-stress components, turbulent kinetic energy (TKE), and dominant wavenumber spectra show good agreement with the reference. Moreover, in comparison with an alternative framework of full-field modeling (i.e., without field decomposition), the proposed framework yields more accurate and robust long-term statistical predictions.

physics.flu-dyn↗

arXiv2026

Enabling High-Accuracy Data Assimilation with Limited Ensembles via Machine Learning-Based Covariance Correction

Data assimilation (DA) integrates numerical model forecasts with observations to achieve the optimal state estimation. Ensemble-based methods, such as the ensemble Kalman filter (EnKF), are widely used for state estimation for high-dimensional and nonlinear dynamic systems. However, their performance strongly depends on the ensemble size, therefore causing a tradeoff problem between analysis accuracy and computational cost. To address this problem, this study presents a machine learning-based EnKF framework that maintains high accuracy with a relatively small ensemble size. Specifically, a multilayer perceptron (MLP) function is built to predict the difference between the forecast error covariances estimated from a limited ensemble and a sufficiently large ensemble, with the latter being assumed to be an accurate approximation of the underlying truth. This predicted covariance difference term is then incorporated into the EnKF algorithm via an element-wise scaling strategy, resulting in an amended forecast covariance matrix that better approximates the true uncertainty level and sequentially produces more accurate analysis results. To demonstrate the feasibility and robustness of the proposed algorithm, we perform a set of numerical experiments with the Lorenz-63 and Lorenz-96 systems under various configurations, and the results consistently indicate that the proposed algorithm can significantly outperform the standard EnKF with the same limited ensemble size, by achieving notably higher analysis accuracy while remaining computationally efficient. This approach provides a practical and feasible pathway to accurate and computationally efficient data assimilation for high-dimensional and nonlinear dynamic systems.

physics.ao-ph↗

arXiv2026

A Novel $\alpha\beta$-Approximation Method Based on Numerical Integration for Discretizing Continuous Systems

In this article, we propose a novel discretization method based on numerical integration for discretizing continuous systems, termed the $\alpha\beta$-approximation or Scalable Bilinear Transformation (SBT). In contrast to existing methods, the proposed method consists of two factors, i.e., shape factor ($\alpha$) and time factor ($\beta$). Depending on the discretization technique applied, we identify two primary distortion modes in discrete resonant controllers: frequency warping and resonance damping. We further provide a theoretical explanation for these distortion modes, and demonstrate that the performance of the method is superior to all typical methods. The proposed method is implemented to discretize a quasi-resonant (QR) controller on a control board, achieving 25\% reduction in the root-mean-square error (RMSE) compared to the SOTA method. Finally, the approach is extended to discretizing a resonant controller of a grid-tied inverter. The efficacy of the proposed method is conclusively validated through favorable comparisons among the theory, simulation, and experiments.

eess.SY↗

arXiv2025

Optimized Design of the Generalized Bilinear Transformation for Discretizing Analog Systems

A common approach to digital system design involves transforming a continuous-time (s-domain) transfer function into the discrete-time (z-domain) using methods such as Euler or Tustin. These transformations are shown to be specific cases of the Generalized Bilinear Transformation (GBT), characterized by a design parameter, $\alpha$, whose physical interpretation and optimal selection remain inadequately explored. In this paper, we propose an alternative derivation of the GBT derived by employing a new hexagonal shape to approximate the enclosed area of the error function, and we define the parameter $\alpha$ as a shape factor. We reveal, for the first time, the physical meaning of $\alpha$ as the backward rectangular ratio of the proposed hexagonal shape. Through domain mapping, the stable range of is rigorously established to be [0.5, 1]. Depending on the operating frequency and the chosen $\alpha$, we observe two distinct distortion modes, i.e., the magnitude and phase distortion. We further develop an optimal design method for $\alpha$ by minimizing a normalized magnitude or phase error objective function. The effectiveness of the proposed method is validated through the design and testing of a low-pass filter (LPF), demonstrating strong agreement between theoretical predictions and experimental results.

eess.SY↗

arXiv2025

Small Ensemble-based Data Assimilation: A Machine Learning-Enhanced Data Assimilation Method with Limited Ensemble Size

Ensemble-based data assimilation (DA) methods have become increasingly popular due to their inherent ability to address nonlinear dynamic problems. However, these methods often face a trade-off between analysis accuracy and computational efficiency, as larger ensemble sizes required for higher accuracy also lead to greater computational cost. In this study, we propose a novel machine learning-based data assimilation approach that combines the traditional ensemble Kalman filter (EnKF) with a fully connected neural network (FCNN). Specifically, our method uses a relatively small ensemble size to generate preliminary yet suboptimal analysis states via EnKF. A FCNN is then employed to learn and predict correction terms for these states, thereby mitigating the performance degradation induced by the limited ensemble size. We evaluate the performance of our proposed EnKF-FCNN method through numerical experiments involving Lorenz systems and nonlinear ocean wave field simulations. The results consistently demonstrate that the new method achieves higher accuracy than traditional EnKF with the same ensemble size, while incurring negligible additional computational cost. Moreover, the EnKF-FCNN method is adaptable to diverse applications through coupling with different models and the use of alternative ensemble-based DA methods.

cs.LG↗

arXiv2024

Ground states of a nonlocal variational problem and Thomas-Fermi limit for the Choquard equation

We study nonnegative optimizers of a Gagliardo-Nirenberg type inequality $$\iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)|^p\,|u(y)|^p}{|x - y|^{N-\alpha}} dx\, dy\le C\Big(\int_{{\mathbb R}^N}|u|^2 dx\Big)^{p\theta} \Big(\int_{{\mathbb R}^N}|u|^q dx\Big)^{2p(1-\theta)/q},$$ that involves the nonlocal Riesz energy with $0<\alpha \frac{N+\alpha}{N}$, $q>\frac{2Np}{N+\alpha}$ and $\theta=\frac{(N+\alpha)q-2Np}{Np(q-2)}$. For $p=2$, the equivalent problem has been studied in connection with the Keller-Segel diffusion-aggregation models in the past few decades. The general case $p\neq 2$ considered here appears in the study of Thomas-Fermi limit regime for the Choquard equations with local repulsion. We establish optimal ranges of parameters for the validity of the above interpolation inequality, discuss the existence and qualitative properties of the nonnegative maximizers, and in some special cases estimate the optimal constant. For $p=2$ it is known that the maximizers are H\"older continuous and compactly supported on a ball. We show that for $p<2$ the maximizers are smooth functions supported on $\mathbb{R}^N$, while for $p>2$ the maximizers consist of a characteristic function of a ball and a nonconstant nonincreasing H\"older continuous function supported on the same ball. We use these qualitative properties of the maximizers to establish the validity of the Thomas-Fermi approximations for the Choquard equations with local repulsion. The results are verified numerically with extensive examples.

math.AP↗

arXiv2022

Asymptotic profile of ground states for the Schr\"{o}dinger-Poisson-Slater equation

We study the Schr\"{o}dinger-Poisson-Slater equation $$-\Delta u + u+\lambda(I_{2}*|u|^2)u=|u|^{p-2}u\quad\text{in $\mathbb R^3$},$$ where $p\in (3,6)$ and $\lambda>0$. By using direct variational analysis based on the comparison of the ground state energy levels, we obtain a characterization of the limit profile of the positive ground states for $\lambda\to \infty$.

math.AP↗

arXiv2021

Limit profiles for singularly perturbed Choquard equations with local repulsion

We study Choquard type equation of the form $$-\Delta u +\varepsilon u-(I_{\alpha}*|u|^p)|u|^{p-2}u+|u|^{q-2}u=0\quad in \quad {\mathbb R}^N,\qquad\qquad(P_\varepsilon)$$ where $N\geq3$, $I_\alpha$ is the Riesz potential with $\alpha\in(0,N)$, $p>1$, $q>2$ and $\varepsilon\ge 0$. Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qualitative properties of positive groundstates of $(P_0)$ and of $(P_\varepsilon)$ with $\varepsilon>0$. We also study the existence of a compactly supported groundstate for an integral Thomas-Fermi type equation associated to $(P_\varepsilon)$. In the second part of the paper, for $\varepsilon\to 0$ we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of $(P_\varepsilon)$ in each of the regimes. We also outline three different asymptotic regimes in the case $\varepsilon\to\infty$. In one of the asymptotic regimes positive groundstates of $(P_\varepsilon)$ converge to a compactly supported Thomas-Fermi limit profile. This is a new and purely nonlocal phenomenon that can not be observed in the local prototype case of $(P_\varepsilon)$ with $\alpha=0$. In particular, this provides a justification for the Thomas-Fermi approximation in astrophysical models of self-gravitating Bose-Einstein condensate.

math.AP↗

arXiv2021

Nonlinear Inequalities with Double Riesz Potentials

We investigate the nonnegative solutions to the nonlinear integral inequality $u \ge I_{\alpha}\ast\big((I_\beta \ast u^p)u^q\big)$ a.e. in $\mathbb{R}^N$, where $\alpha, \beta\in (0,N)$, $p, q>0$ and $I_\alpha$, $I_\beta$ denote the Riesz potentials of order $\alpha$ and $\beta$ respectively. Our approach relies on a nonlocal positivity principle which allows us to derive optimal ranges for the parameters $\alpha$, $\beta$, $p$ and $q$ to describe the existence and the nonexistence of a solution. The optimal decay at infinity for such solutions is also discussed.

math.AP↗

arXiv2020

Epitaxial Growth of $β$-Ga$_2$O$_3$ Coated Wide Bandgap Semiconductor Tape for Flexible UV Photodetector

The epitaxial growth of technically-important $β$-Ga$_2$O$_3$ semiconductor thin films have not been realized on flexible substrates due to limitations by the high-temperature crystallization conditions and the lattice-matching requirements. In this report, for the first time single crystal $β$-Ga$_2$O$_3$(-201) thin films is epitaxially grown on the flexible CeO2 (001)-buffered hastelloy tape. The results indicate that CeO$_2$ (001) has a small bi-axial lattice mismatch with $β$-Ga$_2$O$_3$ (-201), thus inducing a simultaneous double-domain epitaxial growth. Flexible photodetectors are fabricated based on the epitaxial $β$-Ga$_2$O$_3$ coated tapes. Measurements show that the obtained photodetectors have a responsivity of 40 mA/W, with an on/off ratio reaching 1000 under 250 nm incident light and 5 V bias voltage. Such photoelectrical performance is already within the mainstream level of the $β$-Ga$_2$O$_3$ based photodetectors by using the conventional rigid single crystal substrates; and more importantly remained robust against more than 1000 cycles of bending tests. In addition, the epitaxy technique described in the report also paves the way for the fabrication of a wide range of flexible epitaxial film devices that utilize the materials with lattice parameters similar to $β$-Ga$_2$O$_3$, including GaN, AlN and SiC.

physics.app-ph↗

arXiv2016

On a critical Kirchhoff problem in high dimensions

In this paper, we consider the following Kirchhoff problem $$ \left\{\aligned -\bigg(a+b\int_Ω|\nabla u|^2dx\bigg)Δu&= λu^{q-1} + μu^{2^*-1}, &\quad \text{in }Ω, \\ u&>0,&\quad\text{in }Ω,\\ u&=0,&\quad\text{on }\partialΩ, \endaligned \right.\eqno{(\mathcal{P})} $$ where $Ω\subset \bbr^N(N\geq4)$ is a bounded domain, $2\leq q<2^*$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent and $a$, $b$, $λ$, $μ$ are positive parameters. By using the variational method, we obtain some existence and nonexistence results to $(\mathcal{P})$ for all $N\geq4$ with some further conditions on the parameters $a$, $b$, $λ$, $μ$, which partially improve some known results in the literatures. Furthermore, Our result for $N=4$ and $q>2$, together with our previous works \cite{HLW15,HLW151}, gives an almost positive answer to Neimen's open question [J. Differential Equations, 257 (2014), 1168--1193].

math.AP↗

arXiv2016

Positive solutions to an elliptic equation in $\mathbb{R}^N$ of the Kirchhoff type

In this paper, we consider the following Kirchhoff type problem $$\left\{\aligned&-\biggl(a + b\int_{\mathbb{R}^N} |\nabla u|^2 dx \biggr) Δu + V(x) u = |u|^{p-2}u &\text{ in } \mathbb{R}^N,\cr &u\in H^1(\mathbb{R}^N), \endaligned\right. \eqno{(\mathcal{P}_{a,b})} $$ where $N\geq3$, $2 0$ are parameters and $V(x)$ is a potential function. Under some mild conditions on $V(x)$, we prove that $(\mathcal{P}_{a,b})$ has a positive solution for $b$ small enough by the variational method, a non-existence result is also established in the cases $N\geq4$. Our results in the case $N=3$ partial improve the results in \cite{G15,LY14} and our results in the cases $N\geq4$ are totally new to the best of our knowledge. By combining the scaling technique, we also give a global description on the structure of the positive solutions to the autonomous form of $(\mathcal{P}_{a,b})$, that is $V(x)\equivλ>0$. This result can be seen as a partial complement of the studies in \cite{A12,A13}.

math.AP↗

arXiv2015

On finding solutions of a Kirchhoff type problem

Consider the following Kirchhoff type problem $$ \left\{\aligned -\bigg(a+b\int_{\mathbb{B}_R}|\nabla u|^2dx\bigg)Δu&= λu^{q-1} + μu^{p-1}, &\quad \text{in}\mathbb{B}_R, \\ u&>0,&\quad\text{in}\mathbb{B}_R,\\ u&=0,&\quad\text{on}\partial\mathbb{B}_R, \endaligned \right.\eqno{(\mathcal{P})} $$ where $\mathbb{B}_R\subset \bbr^N(N\geq3)$ is a ball, $2\leq q<p\leq2^*:=\frac{2N}{N-2}$ and $a$, $b$, $λ$, $μ$ are positive parameters. By introducing some new ideas and using the well-known results of the problem $(\mathcal{P})$ in the cases of $a=μ=1$ and $b=0$, we obtain some special kinds of solutions to $(\mathcal{P})$ for all $N\geq3$ with precise expressions on the parameters $a$, $b$, $λ$, $μ$, which reveals some new phenomenons of the solutions to the problem $(\mathcal{P})$. It is also worth to point out that it seems to be the first time that the solutions of $(\mathcal{P})$ can be expressed precisely on the parameters $a$, $b$, $λ$, $μ$, and our results in dimension four also give a partial answer to Neimen's open problems [J. Differential Equations, 257 (2014), 1168--1193]. Furthermore, our results in dimension four seems to be almost "optimal".

math.AP↗

arXiv2015

On Kirchhoff type equations with critical Sobolev exponent and Naimen's open problems

We study the following Brezis-Nirenberg problem of Kirchhoff type $$ \left\{\aligned &-(a+b\int_Ω|\nabla u|^2dx)Δu = λ|u|^{q-2}u + δ|u|^{2}u, &\quad \text{in}\ Ω, \\ &u=0,& \text{on}\ \partialΩ, \endaligned \right. $$ where $Ω\subset \bbr^4$ is a bounded domain with the smooth boundary $\partialΩ$, $2\leq q<4$ and $a$, $b$, $λ$, $δ$ are positive parameters. We obtain some new existence and nonexistence results, depending on the values of the above parameters, which improves some known results. The asymptotical behaviors of the solutions are also considered in this paper.

math.AP↗

arXiv2015

On a Kirchhoff type problems with potential well and indefinite potential

In this paper, we study the following Kirchhoff type problem:% $$ \left\{\aligned&-\bigg(α\int_{\bbr^3}|\nabla u|^2dx+1\bigg)Δu+(λa(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{α,λ})}% $$ where $4<p<6$, $α$ and $λ$ are two positive parameters, $a_0\in\bbr$ is a (possibly negative) constant and $a(x)\geq0$ is the potential well. By the variational method, we investigate the existence of nontrivial solutions to $(\mathcal{P}_{α,λ})$. To our best knowledge, it is the first time that the nontrivial solution of the Kirchhoff type problem is found in the indefinite case. We also obtain the concentration behaviors of the solutions as $λ\to+\infty$.

math.AP↗

arXiv2015

On a biharmonic equations with steep potential well and indefinite potential

In this paper, we study the following biharmonic equations:% $$ \left\{\aligned&Δ^2u-a_0Δu+(λb(x)+b_0)u=f(u)&\text{ in }\bbr^N,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_λ)}% $$ where $N\geq3$, $a_0,b_0\in\bbr$ are two constants, $λ>0$ is a parameter, $b(x)\geq0$ is a potential well and $f(t)\in C(\bbr)$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo-Nirenberg inequality, we make some observations on the operator $Δ^2-a_0Δ+λb(x)+b_0$ in $\h$. Based on these observations, we give a new variational setting to $(\mathcal{P}_λ)$ for $a_0<0$. With this new variational setting in hands, we establish some new existence results of the nontrivial solutions to $(\mathcal{P}_λ)$ for all $a_0, b_0\in\bbr$ with $λ$ sufficiently large by the variational method. The concentration behavior of the nontrivial solutions as $λ\to+\infty$ is also obtained. It is worth to point out that it seems to be the first time that the nontrivial solution of $(\mathcal{P}_λ)$ is obtained in the case of $a_0<0$.

math.AP↗