arXiv subjects

Changqing Liu

Publications and source records attributed to Changqing Liu.

At least 19 recordsLinked to original sources

arXiv2026

A Simpler Proof of Kakutani's Conjecture on Random Subdivision and Its Generalizations

Shifting the perspective from division points to division spacings leads not only to a substantially simpler proof of the conjecture but also to stronger results: limiting distributions are obtained for both partition points and spacings, with large-deviation error bounds. Maillard and Paquette's (2016) conjecture \cite{MaillardPaquette2016} is confirmed: the limiting empirical c.d.f. is independent of the splitting scheme. The approach is naturally extended to arbitrary distributions of division points, with the limiting spacing distribution established. It is shown stationarity of the spacing distribution iff stationarity of the mean spacing iff uniform distribution of the partition points. This means that the results of Dean and Majumdar (2002) and Janson and Neininger (2008) concerning the limiting mean spacing apply directly to the limiting spacing distribution. Discrete case (heavy-tailed distribution of division points) is studied and new results are presented, including the stage-wise progression of the fragmentation process. The results obtained apply to random trees.

math.PR↗

arXiv2026

A unified power-grid representation for reuse across network structures and computational tasks

Data-driven power-system models are typically developed for specific grids and computational tasks, but their performance can deteriorate markedly or even fail when network structures or analytical objectives change. This paper develops a unified grid representation that separates physical-grid description from downstream computation. A self-supervised encoder represents each grid as a variable number of fixed-dimensional node, branch and global vectors under a common latent description. Although pretrained only on systems with at most 270 buses, the frozen encoder transfers without adaptation to a 70,000-bus network, more than 250 times larger in bus count, while preserving 0.7919 bus-correspondence accuracy. The same representation supports four independently trained downstream tasks: power-flow calculation, reactive-power adjustment, operating-condition generation and transient-stability assessment. On the previously unseen 70,000-bus grid, power-flow calculation achieves mean errors of 0.108$^{\circ}$ in phase angle and 3.8$\times$10\textsuperscript{-6} p.u. in voltage magnitude. Reactive-power adjustment achieved a 75\% PSASP-verified correction rate on the previously unseen 70,000-bus grid, while operating-condition generation delivered at least one PSASP-verified feasible state for 82.2\% of requests on previously unseen 400--1,000-bus grid families. For transient-stability assessment, the frozen encoder achieves accuracy within 0.8 percentage points of full encoder adaptation under matched training conditions. These results demonstrate that a unified power-grid representation can be learned once and reused unchanged across structurally different power systems and heterogeneous computational tasks.

eess.SY↗

arXiv2026

Concentration Inequalities for Branching Random Walks

Motivated by phase transition problems in CSPs, we prove a more general concentration inequality that retains classical sub-Gaussian tails under a mild global linear-growth condition $|S_n| < Cn$, relaxing the bounded-increment assumption to finite exponential moments and requiring neither independence nor the martingale property. We further extend it to branching random walks (BRWs), obtaining the first concentration inequality for BRWs.

math.PR↗

arXiv2026

Concentration Inequalities for Branching Random Walks with Applications to Phase Transitions in CSPs

A new framework is developed for studying phase transitions in CSPs. Motivated by phase transition problems in CSPs, we prove a more general concentration inequality that retains classical sub-Gaussian tails under a mild global linear-growth condition $|S_n| < Cn$, relaxing the bounded-increment assumption to finite exponential moments and requiring neither independence nor the martingale property. We further extend the concentration inequality to branching random walks (BRW), obtaining the first concentration inequality for BRW. As applications, we derive partial differential equations (PDEs) for the $K$-SAT and $q$-COL backbones, yielding new results, including \textbf{(a)} a resolution of the long-standing open question of where $(2+p)$-SAT transition changes from second to first order; \textbf{(b)} rigorous results for $α_d$ in $K$-SAT, which give new lower bounds on the phase transition for 3-SAT (4.0029 vs. 3.51) and 4-SAT (8.360 vs. 7.91); and \textbf{(c)} the prefactor of the 2-SAT critical window.

cs.CC↗

arXiv2026

Chernoff bounds for branching random walks

Concentration inequalities, which have proved very useful in a variety of fields, provide fairly tight bounds on large deviation probabilities while central limit theorem (CLT) describes the asymptotic distribution around the mean (at the $\sqrt{n}$ scale). Harris (1963) conjectured that for a supercritical branching random walk (BRW) of i.i.d offspring and i.i.d displacements, positions of individuals in $nth$ generation approach to Gaussian distribution -- central limit theorem. This conjecture was later proved by Stam (1966) and Kaplan \& Asmussen (1976). Refinements and extensions followed. However, to the best of our knowledge, there is no corresponding existing work on concentration inequalities for BRWs. In this note, we propose a new definition of BRW, providing a more general framework. Owing to this definition, a Chernoff-type (subgaussian) bound for BRWs follows directly from the Chernoff bound for random walk. The relation between RW (random walk) and BRW is discussed.

math.PR↗

arXiv2026

Exact spectrum of the XX spin chain with constrained non-diagonal boundary fields

We study the exact spectrum of the XX spin chain with constrained non-diagonal boundary fields, which can be analyzed by solving the associated Bethe Ansatz equations. In these equations, the number of Bethe roots has a definite parity, and all Bethe roots are located at the zeros of a unary function. We investigate the possible positions of the Bethe roots. Based on numerical observations, we analyze the Bethe root configurations for the ground state and the first excited state. Our results show that elementary excitations are characterized by the cooperative change of a pair of Bethe roots. Furthermore, we obtain an analytical expression for the ground state energy in the thermodynamic limit.

math-ph↗

arXiv2026

Bivariate partial mapping for detecting causality in complex non-autonomous system

Identifying causality is fundamental for human understanding of the world, where complex non-autonomous systems such as species population changes, brain activities, etc. are extensively existed. Since the phase spaces of such systems are not manifolds, the existing method based on convergent cross mapping is not applicable. This paper proposes a novel bivariate partial mapping method for detecting causality in complex non-autonomous systems. It transforms a non-autonomous system to an autonomous skew product system, and then, by considering the causality changes due to the transformation, detects causality of the original non-autonomous system from the transformed skew product system. The effectiveness of the proposed method is verified by mathematical cases and a real brain activity case, showing that the proposed method successfully detects the causality in complex non-autonomous systems.

math.DS↗

arXiv2026

A scalable neural bundle map for multiphysics prediction in lithium-ion battery across varying configurations

Efficient and accurate prediction of Multiphysics evolution across diverse cell geometries is fundamental to the design, management and safety of lithium-ion batteries. However, existing computational frameworks struggle to capture the coupled electrochemical, thermal, and mechanical dynamics across diverse cell geometries and varying operating conditions. Here, we present a Neural Bundle Map (NBM), a mathematically rigorous framework that reformulates multiphysics evolution as a bundle map over a geometric base manifold. This approach enables the complete decoupling of geometric complexity from underlying physical laws, ensuring strong operator continuity across varying domains. Our framework achieves high-fidelity spatiotemporal predictions with a normalized mean absolute error of less than 1% across varying configurations, while maintaining stability during long-horizon forecasting far beyond the training window and reducing computational costs by two orders of magnitude compared with conventional solvers. Leveraging this capability, we rapidly explored a vast configurational space to identify an optimal battery design that yields a 38% increase in energy density while adhering to thermal safety constraints. Furthermore, the NBM demonstrates remarkable scalability to multi-cell systems through few-shot transfer learning, providing a foundational paradigm for the intelligent design and real-time monitoring of complex energy storage infrastructures.

cs.CE↗

arXiv2025

Neural Diffeomorphic-Neural Operator for Residual Stress-Induced Deformation Prediction

Accurate prediction of machining deformation in structural components is essential for ensuring dimensional precision and reliability. Such deformation often originates from residual stress fields, whose distribution and influence vary significantly with geometric complexity. Conventional numerical methods for modeling the coupling between residual stresses and deformation are computationally expensive, particularly when diverse geometries are considered. Neural operators have recently emerged as a powerful paradigm for efficiently solving partial differential equations, offering notable advantages in accelerating residual stress-deformation analysis. However, their direct application across changing geometric domains faces theoretical and practical limitations. To address this challenge, a novel framework based on diffeomorphic embedding neural operators named neural diffeomorphic-neural operator (NDNO) is introduced. Complex three-dimensional geometries are explicitly mapped to a common reference domain through a diffeomorphic neural network constrained by smoothness and invertibility. The neural operator is then trained on this reference domain, enabling efficient learning of deformation fields induced by residual stresses. Once trained, both the diffeomorphic neural network and the neural operator demonstrate efficient prediction capabilities, allowing rapid adaptation to varying geometries. The proposed method thus provides an effective and computationally efficient solution for deformation prediction in structural components subject to varying geometries. The proposed method is validated to predict both main-direction and multi-direction deformation fields, achieving high accuracy and efficiency across parts with diverse geometries including component types, dimensions and features.

cs.LG↗

arXiv2024

Phantom hairy black holes and wormholes in Einstein-bumblebee gravity

In this paper we study Einstein-bumblebee gravity theory minimally coupled with external matter -- a phantom/non-phantom(conventional) scalar field, and derive a series of hairy solutions -- bumblebee-phantom(BP) and BP-dS/AdS black hole solutions, regular Ellis-bumblebee-phantom (EBP) and BP-AdS wormholes, etc. We first find that the Lorentz violation (LV) effect can change the so called black hole no-hair theorem and these scalar fields can give a hair to a black hole. If LV coupling constant $\ell>-1$, the phantom field is admissible and the conventional scalar field is forbidden; if $\ell<-1$, the phantom field is forbidden and the conventional scalar field is admissible. By defining the Killing potential $ω^{ab}$, we study the Smarr formula and the first law for the BP black hole, find that the appearance of LV can improve the structure of these phantom hairy black holes -- the conventional Smarr formula and the first law of black hole thermodynamics still hold; but for no LV case, i.e., the regular phantom black hole reported in [Phys. Rev. Lett. {\bf96}, 251101], the first law cannot be constructed at all. When the bumblebee potential is linear, we find that the phantom potential and the Lagrange-multiplier $λ$ behave as a cosmological constant $Λ$.

gr-qc↗

arXiv2024

A general reduced-order neural operator for spatio-temporal predictive learning on complex spatial domains

Predictive learning for spatio-temporal processes (PL-STP) on complex spatial domains plays a critical role in various scientific and engineering fields, with its essence being the construction of operators between infinite-dimensional function spaces. This paper focuses on the unequal-domain mappings in PL-STP and categorising them into increase-domain and decrease-domain mapping. Recent advances in deep learning have revealed the great potential of neural operators (NOs) to learn operators directly from observational data. However, existing NOs require input space and output space to be the same domain, which pose challenges in ensuring predictive accuracy and stability for unequal-domain mappings. To this end, this study presents a general reduced-order neural operator named Reduced-Order Neural Operator on Riemannian Manifolds (RO-NORM), which consists of two parts: the unequal-domain encoder/decoder and the same-domain approximator. Motivated by the variable separation in classical modal decomposition, the unequal-domain encoder/decoder uses the pre-computed bases to reformulate the spatio-temporal function as a sum of products between spatial (or temporal) bases and corresponding temporally (or spatially) distributed weight functions, thus the original unequal-domain mapping can be converted into a same-domain mapping. Consequently, the same-domain approximator NORM is applied to model the transformed mapping. The performance of our proposed method has been evaluated on six benchmark cases, including parametric PDEs, engineering and biomedical applications, and compared with four baseline algorithms: DeepONet, POD-DeepONet, PCA-Net, and vanilla NORM. The experimental results demonstrate the superiority of RO-NORM in prediction accuracy and training efficiency for PL-STP.

cs.LG↗

arXiv2024

Diffeomorphism Neural Operator for various domains and parameters of partial differential equations

In scientific and engineering applications, solving partial differential equations (PDEs) across various parameters and domains normally relies on resource-intensive numerical methods. Neural operators based on deep learning offered a promising alternative to PDEs solving by directly learning physical laws from data. However, the current neural operator methods were limited to solve PDEs on fixed domains. Expanding neural operators to solve PDEs on various domains hold significant promise in medical imaging, engineering design and manufacturing applications, where geometric and parameter changes are essential. This paper presents a novel neural operator learning framework for solving PDEs with various domains and parameters defined for physical systems, named diffeomorphism neural operator (DNO). The main idea is that a neural operator learns in a generic domain which is diffeomorphically mapped from various physics domains expressed by the same PDE. In this way, the challenge of operator learning on various domains is transformed into operator learning on the generic domain. The generalization performance of DNO on different domains can be assessed by a proposed method which evaluates the geometric similarity between a new domain and the domains of training dataset after diffeomorphism. Experiments on Darcy flow, pipe flow, airfoil flow and mechanics were carried out, where harmonic and volume parameterization were used as the diffeomorphism for 2D and 3D domains. The DNO framework demonstrated robust learning capabilities and strong generalization performance across various domains and parameters.

math.NA↗

arXiv2023

Learning Neural Operators on Riemannian Manifolds

In Artificial Intelligence (AI) and computational science, learning the mappings between functions (called operators) defined on complex computational domains is a common theoretical challenge. Recently, Neural Operator emerged as a promising framework with a discretisation-independent model structure to break the fixed-dimension limitation of classical deep learning models. However, existing operator learning methods mainly focus on regular computational domains, and many components of these methods rely on Euclidean structural data. In real-life applications, many operator learning problems are related to complex computational domains such as complex surfaces and solids, which are non-Euclidean and widely referred to as Riemannian manifolds. Here, we report a new concept, Neural Operator on Riemannian Manifolds (NORM), which generalises Neural Operator from being limited to Euclidean spaces to being applicable to Riemannian manifolds, and can learn the mapping between functions defined on any real-life complex geometries, while preserving the discretisation-independent model structure. NORM shifts the function-to-function mapping to finite-dimensional mapping in the Laplacian eigenfunctions' subspace of geometry, and holds universal approximation property in learning operators on Riemannian manifolds even with only one fundamental block. The theoretical and experimental analysis prove that NORM is a significant step forward in operator learning and has the potential to solve complex problems in many fields of applications sharing the same nature and theoretical principle.

math.NA↗

arXiv2023

Rotating BTZ-like black hole and central charges in Einstein-bumblebee gravity

We obtain an exact rotating BTZ-like black hole solution by solving the corresponding gravitational field equations and the bumblebee motion equations in Einstein-bumblebee gravity theory. Result is presented for the purely radial Lorentz symmetry violating and can only exist with a linear functional potential of the bumblebee field. This black hole has two horizons and an ergosphere which are dependent on the bumblebee coupling constant $\ell$. The concepts of the area and volume of the horizon should be renewed in this LV spacetime due to the nontrivial contribution of coupling between the bumblebee field and the Ricci tensor. Only in this way, the entropy-area relation, first law of thermodynamics and the Smarr formula can still be constructed. We also study the AdS/CFT correspondence of this black hole, find that the entropy product of its inner and outer horizons is universal. So the central charges of the dual CFT on the boundary can be obtained via the thermodynamic method, and they can reappear black hole mass and angular momentum in the bulk.

gr-qc↗

arXiv2022

High dimensional AdS-like black hole and Phase transition in Einstein-bumblebee gravity

In this paper we obtain an exact high dimensional anti-de Sitter (AdS) black hole solution in Einstein-bumblebee gravity theory. This AdS-like black hole can only exist with a linear functional potential of the bumblebee field. We find that the Smarr formula and the first law of black hole thermodynamics can still be constructed in this Lorentz symmetry breaking black hole spacetime as long as its temperature, entropy and volume are slightly modified. We find also that there exist two kinds of phase transition: small-large black hole phase transition and Hawking-Page phase transition, like those of Schwarzschild AdS black hole. After Lorentz symmetry breaking, the black hole mass at divergent point of heat capacity becomes small, and the Gibbs free energy of the meta-stable large black hole is also smaller, showing that the large stable black hole can be more easily formed.

gr-qc↗

arXiv2021

Image of the Schwarzschild black hole pierced by a cosmic string with a thin accretion disk

We study the optical appearance of a thin accretion disk around a Schwarzschild black hole pierced by a cosmic string with a semi-analytic method of Luminet [11]. Direct and secondary images with different parameters observed by a distant observer is plotted. The cosmic string parameter s can modify the shape and size of the thin disk image. We calculate and plot the distribution of both redshift and observed flux as seen by distant observers at different inclination angles. Those distributions are dependent on the inclination angel of the observer and cosmic parameter s.

gr-qc↗

arXiv2020

Exact Kerr-like solution and its shadow in a gravity model with spontaneous Lorentz symmetry breaking

We obtain an exact Kerr-like black hole solution by solving the corresponding gravitational field equations in Einstein-bumblebee gravity model where Lorentz symmetry is spontaneously broken once a vector field acquires a vacuum expectation value. Results are presented for the purely radial Lorentz symmetry breaking. In order to study the effects of this breaking, we consider the black hole shadow and find that the radial of the unstable spherical orbit on the equatorial plane $r_c$ decreases with the Lorentz breaking constant $\ell>0$, and increases with $\ell<0$. These shifts are similar to those of Einstein-aether black hole. The effect of the LV parameter on the black hole shadow is that it accelerates the appearance of shadow distortion, and could be detected by the new generation of gravitational antennas.

gr-qc↗

arXiv2019

Periodic orbits around Kerr Sen black holes

We investigate periodic orbits and zoom-whirl behaviors around a Kerr Sen black hole with a rational number $q$ in terms of three integers $(z,w,v)$, from which one can immediately read off the number of leaves(or zooms), the ordering of the leaves, and the number of whirls. The characteristic of zoom-whirl periodic orbits is the precession of multi-leaf orbits in the strong field regime. This feature is analogous to the counterpart in the Kerr space-time. Finally, we analyze the impact of the charge parameter $b$ on the zoom-whirl periodic orbits. Compared to the periodic orbits around the Kerr black hole, it is found that typically lower energies are required for the same orbits in the Kerr Sen black hole.

gr-qc↗