arXiv subjects

Xiaofei Zhao

Publications and source records attributed to Xiaofei Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Some uniform error analysis for gKdV equations in dispersionless limit regime before dispersive shock

This work establishes uniform error estimates for classical numerical schemes applied to the generalized Korteweg-de Vries (gKdV) equation in the dispersionless limit regime, specifically before the development of dispersive shock wave (DSW). We analyze the Crank-Nicolson (CN) and a Lawson-type Runge-Kutta (Lawson-RK) methods, when discretized in space via the Fourier pseudo-spectral method. We prove that both fully discrete schemes achieve optimal second-order temporal accuracy and spectral spatial accuracy, with error constants being uniform in the vanishing dispersion parameter $\varepsilon$. The analysis also addresses the unique solvability of the CN scheme in the dispersionless limit regime. Theoretical findings are supported by numerical experiments, demonstrating the $\varepsilon$-uniform accuracy and the necessity of step size restriction for CN. The study in general validates the classical methods for solving the dispersionless gKdV equation before DSW.

math.NA↗

arXiv2026

Multi-stage neural operator learning with application for convolutions

Convolution integrals widely exist in applications, and to enable fast and accurate computations, this paper introduces two general multi-stage neural operator learning frameworks. The first, Deep Collocation Neural Operator (DCNO), is a supervised approach that iteratively refines the operator approximation by learning residuals from input-output data pairs. The second, Deep Galerkin Neural Operator (DGNO), is an unsupervised framework applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training. Both methods progressively construct basis operators through multiple training stages to enrich the approximation space, leading to significantly improved accuracy over standard one-shot operator learning. We provide theoretical analysis for their approximation capabilities and implement them for learning convolutions. Extensive numerical experiments demonstrate that both DCNO and DGNO achieve high accuracy, approaching machine precision under single float for convolution problems, and offer substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers. We also extend these frameworks to handle multi-input operator learning scenarios involving variations in both the density and kernel of a convolution.

cs.LG↗

arXiv2026

Nonlocal modeling of opinion alignment and environmental feedback: Spatial aggregation and non-consensus patterns

The formation of public opinion in modern information environments is shaped by the interplay between social conformity and information exposure. While social interactions promote opinion alignment, heterogeneous visibility and selective exposure may reinforce local agreement, a mechanism commonly associated with the echo chamber effects. To describe how such reinforcement influences spatially heterogeneous opinion activity and non-consensus patterns, we propose a spatial opinion dynamics model with attention-mediated feedback. The model couples nonlocal alignment with an evolving attention field and captures a self-reinforcing mechanism in which regions of high opinion activity attract greater visibility. Starting from agent-based jump mechanism inspired by bounded confidence interactions and biased random walks induced by environments, we formally derive a nonlocal advection-cross-diffusion system, where opinion transport is driven by nonlocal conformity and modulated by attention-dependent redistribution. We characterize the transition from spatially homogeneous consensus states to non-consensus clustered regimes through linear stability analysis of the homogeneous equilibrium. The results show that attention-mediated feedback has an explicit correction on the instability threshold and enlarges the parameter regime in which clustering occurs, thereby promoting persistent spatial heterogeneity and non-consensus patterns. Numerical simulations based on a structure-preserving IMEX spectral method support the theoretical predictions and quantify the resulting aggregation phenomena. These findings provide a macroscopic description of how nonlocal alignment and environmental feedback jointly shape spatial signatures of non-consensus patterns.

math.NA↗

arXiv2026

Numerical solution of the nonlinear Dirac equation by a splitting variational quantum algorithm

In this work, we propose an operator-splitting variational quantum algorithm, termed Dirac-sVQA, for simulating the nonlinear Dirac equation (NLDE). The main difficulty arises from the state-dependent nonlinear interaction, its time-discrete update depends explicitly on the intermediate spinor state and, in general, cannot be implemented as a fixed state-independent unitary circuit. To address this difficulty, we decompose the NLDE evolution into a structured linear Dirac substep and a nonlinear variational correction. The linear substep is implemented by a spinor-Fourier Dirac propagator on a joint position-spin register, preserving the spin-momentum coupling and mass-induced spin evolution of the Dirac operator. The nonlinear correction is reformulated as a measurement-based variational update through a small set of overlap, self-channel, and cross-channel observables. We provide the corresponding quantum circuits and derive measurement-aware resource and complexity estimates. Numerical experiments in several nonlinear regimes show that Dirac-sVQA accurately captures both the total density and the componentwise spinor dynamics, agrees well with classical Fourier pseudospectral splitting solutions, and exhibits stable error behavior over time. These results provide numerical evidence for the feasibility of operator-splitting variational quantum simulation for nonlinear relativistic wave equations.

quant-ph↗

arXiv2026

Analysis of gradient flow for computing defocusing action ground states of rotating nonlinear Schr\"odinger equations

This work focuses on the numerical computation of defocusing action ground states for rotating nonlinear Schr\"odinger equations (RNLS) using a direct gradient flow (DGF) method. We address theoretical gaps in the existing literature concerning the stability and convergence of this DGF scheme. Firstly, we prove the unconditional stability of the DGF scheme, demonstrating that the action functional is monotonically non-increasing along the discrete flow for arbitrary time step sizes. Secondly, we establish a rigorous convergence analysis, proving global convergence under minor assumptions and local exponential convergence to the action ground state under a reasonable non-degeneracy condition. The analysis relies on the uniform boundedness of sublevel sets of the action functional and introduces a tailored $H^1$-distance between phase-shift equivalence classes to handle complex-valued ground states with quantized vortices. A novel analytical framework is also developed to establish the exponential convergence rate. Numerical experiments are presented to validate the theoretical findings, demonstrating both the global migration towards a neighborhood of the ground state and subsequent exponential convergence.

math.NA↗

arXiv2026

Convergence analysis of $L^{p+1}$-normalized gradient flow for action ground state of nonlinear Schr\"odinger equation

This paper presents a rigorous convergence analysis of the $L^{p+1}$-normalized gradient flow with asymptotic Lagrange multiplier (GFALM) method for computing the action ground state of the nonlinear Schr\"odinger equation in the focusing case. First, a general global convergence theory is established for the semi-discrete GFALM scheme, guaranteeing the existence of an accumulation point and a convergent subsequence. Then, under additional non-degeneracy assumptions, a local exponential convergence rate is rigorously proven. This result is further extended to the fully discrete case using a Fourier pseudo-spectral discretization. The analysis is achieved by characterizing the local geometry of the $L^{p+1}$-constrained manifold near the ground state, establishing a quadratic growth property of the energy functional, and deriving a \L{}ojasiewicz-type gradient inequality. Finally, the paper also investigates the exponential convergence of the associated continuous-time gradient flow, providing a theoretical foundation for future numerical discretizations. This work extends existing convergence analyses for energy ground states, addressing the challenges posed by the $L^{p+1}$ constraint, especially the absence of an inner-product structure.

math.NA↗

arXiv2025

Feeding a Kerr black hole with quantized vortices

By solving a nonlinear Klein-Gordon equation in Kerr geometry, we uncover new phenomena and key characteristics of quantized vortices in quantum fluids near a Kerr black hole. The formation of these vortices induces rotational or turbulent flows, which profoundly alter the fluid properties and revise those dark matter models describing axion condensates, ultralight boson clouds, and other scalar fields in the vicinity of spinning black holes. As macroscopic, quantum, and topological defects, these vortices can stably orbit the black hole over extended periods, establishing their viability as novel probes for investigating black hole physics. For instance, we calculate the angular velocities of orbiting vortices to quantitatively characterize the frame-dragging effect, a classic prediction of general relativity. Additionally, we observe that relatively large vortices are accreted onto the black hole, wrapping around it while undergoing splitting and reconnecting processes. In quantum fluids with high vortex densities, turbulent flows emerge, accompanied by the formation of a vortex boundary layer near the event horizon. Beyond the ergosphere, we find vortex emissions and energetic outbursts, which may provide crucial insights into analogous astrophysical events recently discovered by the XRISM satellite.

gr-qc↗

arXiv2025

Toward fast, accurate and robust AI prediction of ground states in rotating BEC

We propose an unsupervised deep learning approach for computing the ground state (GS) of rotating Bose-Einstein condensation. To minimize the energy under a mass constraint, our approach introduces two key and novel ingredients: a normalized loss function that exactly enforces the mass constraint, and a training strategy named virtual rotation acceleration that is essential for avoiding local minima and guiding the learning process to the correct quantized vortex phase. Extensive numerical experiments demonstrate the proposed approach as an effective and accurate method to predict GS across physical conditions--from slow to fast rotation and from isotropic to anisotropic confinement. Through further distillation, we establish a unified operator network capable of efficiently generalizing physical parameters across different phases. It enables rapid GS predictions while correctly capturing phase transitions and is applied for inverse problems.

cond-mat.quant-gas↗

arXiv2025

Quasi-Monte Carlo time-splitting methods for the Schr\"odinger equation with Gaussian random potential

In this paper, we study the Schr\"odinger equation with a Gaussian random potential (SE-GP) and develop an efficient numerical method to approximate the expectation of physical observables. The unboundedness of Gaussian random variables poses significant difficulties in both sampling and error analysis. Under time-splitting discretizations of SE-GP, we establish the regularity of the semi-discrete solution in the random space. Then, we introduce a non-standard weighted Sobolev space with properly chosen weight functions, and obtain a randomly shifted lattice-based quasi-Monte Carlo (QMC) quadrature rule for efficient sampling. This approach leads to a QMC time-splitting (QMC-TS) scheme for solving the SE-GP. We prove that the proposed QMC-TS method achieves a dimension-independent convergence rate that is almost linear with respect to the number of QMC samples. Numerical experiments illustrate the sharpness of the error estimate.

math.NA↗

arXiv2025

Unsupervised operator learning approach for dissipative equations via Onsager principle

Existing operator learning methods rely on supervised training with high-fidelity simulation data, introducing significant computational cost. In this work, we propose the deep Onsager operator learning (DOOL) method, a novel unsupervised framework for solving dissipative equations. Rooted in the Onsager variational principle (OVP), DOOL trains a deep operator network by directly minimizing the OVP-defined Rayleighian functional, requiring no labeled data, and then proceeds in time explicitly through conservation/change laws for the solution. Another key innovation here lies in the spatiotemporal decoupling strategy: the operator's trunk network processes spatial coordinates exclusively, thereby enhancing training efficiency, while integrated external time stepping enables temporal extrapolation. Numerical experiments on typical dissipative equations validate the effectiveness of the DOOL method, and systematic comparisons with supervised DeepONet and MIONet demonstrate its enhanced performance. Extensions are made to cover the second-order wave models with dissipation that do not directly follow OVP.

cs.LG↗

arXiv2025

Arbitrary High-Order Maximum Principle-Preserving and Energy Dissipating Schemes for Gradient Flows

For gradient flows, the existing structure-preserving schemes are difficult to achieve arbitrary high-order accuracy in time while preserving maximum-principle (MBP) and energy dissipating simultaneously. In this paper, we develop a new framework for constructing structure-preserving schemes which shall preserve those nice properties. By introducing KKT-conditions for energy dissipating and bound-preserving, we rewrite the original gradient flow into an expanded and coupled system. We shall utilize a novel predictor-corrector-corrector framework, termed the PCC method, which consists of a prediction from any numerical scheme to the user's favor, followed by two correction steps designed to enforce energy stability and MBP, respectively. We take the exponential time differencing Runge-Kutta scheme (ETDRK) as an example and establish the unique solvability and robust error analysis for our new framework. Extensive numerical experiments are provided to validate the efficiency and accuracy of our new approach. Enough numerical comparisons with the existing popular schemes are shown that our structure-preserving schemes can avoid numerical oscillations and capture the exact evolution of energy.

math.NA↗

arXiv2025

On blowup solution in NLS equation under dispersion or nonlinearity management

In this paper, we study the dispersion-managed nonlinear Schr\"odinger (DM-NLS) equation $$ i\partial_t u(t,x)+\gamma(t)\Delta u(t,x)=|u(t,x)|^{\frac4d}u(t,x),\quad x\in\R^d, $$ and the nonlinearity-managed NLS (NM-NLS) equation: $$ i\partial_t u(t,x)+\Delta u(t,x)=\gamma(t)|u(t,x)|^{\frac4d}u(t,x), \quad x\in\R^d, $$ where $\gamma(t)$ is a periodic function which is equal to $-1$ when $t\in (0,1]$ and is equal to $1$ when $t\in (1,2]$. The two models share the feature that the focusing and defocusing effects convert periodically. For the classical focusing NLS, it is known that the initial data $$ u_0(x)=T^{-\frac{d}{2}}\fe^{i\frac{|x|^2}{4T} -i\frac{\omega^2}{T}}Q_\omega\left(\frac{x}{T}\right) $$ leads to a blowup solution $$(T-t)^{-\frac{d}{2}}\fe^{i\frac{|x|^2}{4(T-t)} -i\frac{\omega^2}{T-t}}Q_\omega\left(\frac{x}{T-t}\right), $$ so when $T\leq1$, this is also a blowup solution for DM-NLS and NM-NLS which blows up in the first focusing layer. For DM-NLS, we prove that when $T>1$, the initial data $u_0$ above does not lead to a finite-time blowup and the corresponding solution is globally well-posed. For NM-NLS, we prove the global well-posedness for $T\in(1,2)$ and we construct solution that can blow up at any focusing layer. The theoretical studies are complemented by extensive numerical explorations towards understanding the stabilization effects in the two models and addressing their difference.

math.AP↗

arXiv2025

Error estimates of time-splitting schemes for nonlinear Klein--Gordon equation with rough data

In this work, we consider the convergence analysis of time-splitting schemes for the nonlinear Klein--Gordon/wave equation under rough initial data. The optimal error bounds of the Lie splitting and the Strang splitting are established with sharp dependence on the regularity index of the solution from a wide range that is approaching the lower bound for well-posedness. Particularly for very rough data, the technique of discrete Bourgain space is utilized and developed, which can apply for general second-order wave models. Numerical verifications are provided.

math.NA↗

arXiv2024

Computing ground states of Bose-Einstein condensation by normalized deep neural network

We propose a normalized deep neural network (norm-DNN) for computing ground states of Bose-Einstein condensation (BEC) via the minimization of the Gross-Pitaevskii energy functional under unitary mass normalization. Compared with the traditional deep neural network for solving partial differential equations, two additional layers are added in training our norm-DNN for solving this kind of unitary constraint minimization problems: (i) a normalization layer is introduced to enforce the unitary mass normalization, and (ii) a shift layer is added to guide the training to non-negative ground state. The proposed norm-DNN gives rise to an efficient unsupervised approach for learning ground states of BEC. Systematical investigations are first carried out through extensive numerical experiments for computing ground states of BEC in one dimension. Extensions to high dimensions and multi-component are then studied in details. The results demonstrate the effectiveness and efficiency of norm-DNN for learning ground states of BEC. Finally, we extend the norm-DNN for computing the first excited states of BEC and discuss parameter generalization issues as well as compare with some existing machine learning methods for computing ground states of BEC in the literature.

cond-mat.quant-gas↗

arXiv2024

Improved scalar auxiliary variable schemes for original energy stability of gradient flows

Scalar auxiliary variable (SAV) methods are a class of linear schemes for solving gradient flows that are known for the stability of a `modified' energy. In this paper, we propose an improved SAV (iSAV) scheme that not only retains the complete linearity but also ensures rigorously the stability of the original energy. The convergence and optimal error bound are rigorously established for the iSAV scheme and discussions are made for its high-order extension. Extensive numerical experiments are done to validate the convergence, robustness and energy stability of iSAV, and some comparisons are made.

math.NA↗

arXiv2024

The cubic nonlinear Schrödinger equation with rough potential

We consider the cubic nonlinear Schrödinger equation with a spatially rough potential, a key equation in the mathematical setup for nonlinear Anderson localization. Our study comprises two main parts: new optimal results on the well-posedness analysis on the PDE level, and subsequently a new efficient numerical method, its convergence analysis and simulations that illustrate our analytical results. In the analysis part, our results focus on understanding how the regularity of the solution is influenced by the regularity of the potential, where we provide quantitative and explicit characterizations. Ill-posedness results are also established to demonstrate the sharpness of the obtained regularity characterizations and to indicate the minimum regularity required from the potential for the NLS to be solvable. Building upon the obtained regularity results, we design an appropriate numerical discretization for the model and establish its convergence with an optimal error bound. The numerical experiments in the end not only verify the theoretical regularity results, but also confirm the established convergence rate of the proposed scheme. Additionally, a comparison with other existing schemes is conducted to demonstrate the better accuracy of our new scheme in the case of a rough potential.

math.NA↗

arXiv2024

Fourier pseudospectral methods for the spatial variable-order fractional wave equations

In this paper, we propose Fourier pseudospectral methods to solve the variable-order space fractional wave equation and develop an accelerated matrix-free approach for its effective implementation. In constant-order cases, our methods can be efficiently implemented via the (inverse) fast Fourier transforms, and the computational cost at each time step is ${\mathcal O}(N\log N)$ with $N$ the total number of spatial points. However, this fast algorithm fails in the variable-order cases due to the spatial dependence of the Fourier multiplier. On the other hand, the direct matrix-vector multiplication approach becomes impractical due to excessive memory requirements. To address this challenge, we proposed an accelerated matrix-free approach for the efficient computation of variable-order cases. The computational cost is ${\mathcal O}(MN\log N)$ and storage cost ${\mathcal O}(MN)$, where $M \ll N$. Moreover, our method can be easily parallelized to further enhance its efficiency. Numerical studies show that our methods are effective in solving the variable-order space fractional wave equations, especially in high-dimensional cases. Wave propagation in heterogeneous media is studied in comparison to homogeneous counterparts. We find that wave dynamics in fractional cases become more intricate due to nonlocal interactions. Specifically, dynamics in heterogeneous media are more complex than those in homogeneous media.

math.NA↗

arXiv2023

Error estimate of a quasi-Monte Carlo time-splitting pseudospectral method for nonlinear Schrodinger equation with random potentials

In this paper, we consider the numerical solution of a nonlinear Schrodinger equation with spatial random potential. The randomly shifted quasi-Monte Carlo (QMC) lattice rule combined with the time-splitting pseudospectral discretization is applied and analyzed. The nonlinearity in the equation induces difficulties in estimating the regularity of the solution in random space. By the technique of weighted Sobolev space, we identify the possible weights and show the existence of QMC that converges optimally at the almost-linear rate without dependence on dimensions. The full error estimate of the scheme is established. We present numerical results to verify the accuracy and investigate the wave propagation.

math.NA↗