arXiv subjects

Zhenhua Shi

Publications and source records attributed to Zhenhua Shi.

12 recordsLinked to original sources

arXiv2026

A Rigorous Proof of Metric Blow-Up for Pseudospherical Metrics Associated with the Degasperis--Procesi Equation

This paper studies finite-time blow-up of pseudospherical metrics induced by solutions of the Cauchy problem for the Degasperis--Procesi equation. For initial momentum profiles satisfying suitable left--right sign conditions, we use the method of characteristics, the momentum-transport formula, the Green's-function representation, and Riccati-type differential inequalities to analyze the metric along a distinguished characteristic. We prove that the associated coframe remains non-degenerate before the critical time, so that the induced pseudospherical metric is well defined in the precritical region. Moreover, as wave breaking is approached, the metric component $g_{22}$ diverges to $+\infty$; when $μ\neq0$, the mixed component $g_{12}$ also blows up in absolute value. Thus, finite-time wave breaking for the Degasperis--Procesi equation is shown to induce blow-up of certain components of the corresponding pseudospherical metric.

math-ph↗

arXiv2026

Systems of partial differential equations describing pseudo-spherical or spherical surfaces

In this paper, we study systems of nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness condition of connection 1-forms, we present a classification of systems of Camassa-Holm-type equations of the form \begin{equation*} \left\{ \begin{aligned} u_{t} - u_{xxt} &= F(x, t, u, u_{x}, \dots, \partial ^{m} u/\partial _x^{m}, v, v_{x}, \dots, \partial ^{n} v/\partial _x^{n}), \\ v_{t} - v_{xxt} &= G(x, t, u, u_{x}, \dots, \partial ^{m} u/\partial _x^{m}, v, v_{x}, \dots, \partial ^{n} v/\partial _x^{n}), \end{aligned} \right. \end{equation*} with $m,n\geq2$, for $F$ and $G$ smooth functions, describing pseudospherical or spherical surfaces. We also establish classification results for a special type of third-order system. Applications of the results provide new examples of such systems, such as the Song-Qu-Qiao system, the two-component Camassa-Holm system with cubic nonlinearity, and the modified Camass-Holm-type system. Moreover, we construct the nonlocal symmetry and a non-trivial solutions for the two-component Camassa-Holm system with cubic nonlinearity from the gradients of spectral parameters.

math-ph↗

arXiv2025

On a class of third order differential equations describing pseudospherical or spherical surfaces

In this paper, we study third order nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness of connection 1-forms, we present a classification of equations with the type $u_t - u_{xxt} = λu^2 u_{xxx} + G(u, u_x, u_{xx}) (λ\in\mathbb{R})$, which describe pseudospherical or spherical surfaces. We show that series of typical soliton equations belong to certain subclass, such as the generalized Camassa-Holm equation, which gives a geometric explanation to these equations.

math-ph↗

arXiv2025

Local isometric immersions of pseudospherical surfaces described by a class of third order differential equations

We discuss a specific type of pseudospherical surfaces defined by a class of third order differential equations, of the form $u_t - u_{xxt} = λu^2 u_{xxx} + G(u, u_x, u_{xx})$, and poses a question about the dependence of the triples $\{a,b,c\}$ of the second fundamental form in the context of local isometric immersion in $\mathbb{E}^3$. It is demonstrated that the triples $\{a,b,c\}$ of the second fundamental form are not influenced by a jet of finite order of $u$. Instead, they are shown to rely on a jet of order zero, making them universal and not reliant on the specific solution chosen for $u$.

math.DG↗

arXiv2025

Local isometric immersions of pseudospherical surfaces described by a class of third order partial differential equations

In this paper, we study the problem of local isometric immersion of pseudospherical surfaces determined by the solutions of a class of third order nonlinear partial differential equations with the type $u_t - u_{xxt} = λu^2 u_{xxx} + G(u, u_x, u_{xx}),(λ\in\mathbb{R})$. We prove that there is only two subclasses of equations admitting a local isometric immersion into the three dimensional Euclidean space $\mathbb{E}^3$ for which the coefficients of the second fundamental form depend on a jet of finite order of $u$, and furthermore, these coefficients are universal, namely, they are functions of $x$ and $t$, independent of $u$. Finally, we show that the generalized Camassa-Holm equation describing pseudospherical surfaces has a universal second fundamental form.

math-ph↗

arXiv2024

Pseudo-potentials and local isometric immersion for a generalized Camassa-Holm equation describing pseudospherical surfaces

A generalized Camassa-Holm equation, which describes pseudospherical surfaces, is considered. Using geometric methods, it is demonstrated that the equation is geometrically integrable. Additionally, an infinite hierarchy of conservation laws is derived. Furthermore, the paper delves into the investigation of the problem of locally isometric immersion into three-dimensional Euclidean space based on the equation.

math-ph↗

arXiv2024

The Quasi-Integrability of a Generalized Camassa-Holm Equation

This paper examines a generalization of the Camassa-Holm equation from the perspective of integrability. Using the framework developed by Dubrovin on bi-Hamiltonian deformations and the general theory of quasi-integrability, we demonstrate that a unique bi-Hamiltonian structure is possible for this generalized equation only when it reduces to the original CH equation.

nlin.SI↗

arXiv2022

BoostTree and BoostForest for Ensemble Learning

Bootstrap aggregating (Bagging) and boosting are two popular ensemble learning approaches, which combine multiple base learners to generate a composite model for more accurate and more reliable performance. They have been widely used in biology, engineering, healthcare, etc. This paper proposes BoostForest, which is an ensemble learning approach using BoostTree as base learners and can be used for both classification and regression. BoostTree constructs a tree model by gradient boosting. It increases the randomness (diversity) by drawing the cut-points randomly at node splitting. BoostForest further increases the randomness by bootstrapping the training data in constructing different BoostTrees. BoostForest generally outperformed four classical ensemble learning approaches (Random Forest, Extra-Trees, XGBoost and LightGBM) on 35 classification and regression datasets. Remarkably, BoostForest tunes its parameters by simply sampling them randomly from a parameter pool, which can be easily specified, and its ensemble learning framework can also be used to combine many other base learners.

cs.LG↗

arXiv2021

An adaptive recursive sliding mode attitude control for tiltrotor UAV in flight mode transition based on super-twisting extended state observer

With the characteristics of vertical take-off and landing and long endurance, tiltrotor has attracted considerable attention in recent decades for its potential applications in civil and scientific research. However, the problems of strong couplings, nonlinear characteristics and mismatched disturbances inevitably exist in the tiltrotor, which bring great challenges to the controller design in transition mode. In this paper, we combined a super-twisting extended state observer (STESO) with an adaptive recursive sliding mode control (ARSMC) together to design a tiltrotor aircraft attitude system controller in transition mode using STESO-ARSMC (SAC). Firstly, the six degrees of freedom (DOF) nonlinear mathematical model of tiltrotor is established. Secondly, the states and disturbances are estimated by the STES observer. Thirdly, ARSM controller is designed to achieve finite time convergence. The Lyapunov function is used to testify the convergence of the tiltrotor UAV system. The new aspect is that the assessments of the states are incorporated into the control rules to adjust for disruptions. When compared to prior techniques, the control system proposed in this work can considerably enhance anti-disturbance performance. Finally, simulation tests are used to demonstrate the efficacy of the suggested technique.

cs.RO↗

arXiv2020

FCM-RDpA: TSK Fuzzy Regression Model Construction Using Fuzzy C-Means Clustering, Regularization, DropRule, and Powerball AdaBelief

To effectively optimize Takagi-Sugeno-Kang (TSK) fuzzy systems for regression problems, a mini-batch gradient descent with regularization, DropRule, and AdaBound (MBGD-RDA) algorithm was recently proposed. This paper further proposes FCM-RDpA, which improves MBGD-RDA by replacing the grid partition approach in rule initialization by fuzzy c-means clustering, and AdaBound by Powerball AdaBelief, which integrates recently proposed Powerball gradient and AdaBelief to further expedite and stabilize parameter optimization. Extensive experiments on 22 regression datasets with various sizes and dimensionalities validated the superiority of FCM-RDpA over MBGD-RDA, especially when the feature dimensionality is higher. We also propose an additional approach, FCM-RDpAx, that further improves FCM-RDpA by using augmented features in both the antecedents and consequents of the rules.

cs.LG↗

arXiv2020

Multi-View Broad Learning System for Primate Oculomotor Decision Decoding

Multi-view learning improves the learning performance by utilizing multi-view data: data collected from multiple sources, or feature sets extracted from the same data source. This approach is suitable for primate brain state decoding using cortical neural signals. This is because the complementary components of simultaneously recorded neural signals, local field potentials (LFPs) and action potentials (spikes), can be treated as two views. In this paper, we extended broad learning system (BLS), a recently proposed wide neural network architecture, from single-view learning to multi-view learning, and validated its performance in decoding monkeys' oculomotor decision from medial frontal LFPs and spikes. We demonstrated that medial frontal LFPs and spikes in non-human primate do contain complementary information about the oculomotor decision, and that the proposed multi-view BLS is a more effective approach for decoding the oculomotor decision than several classical and state-of-the-art single-view and multi-view learning approaches.

q-bio.NC↗

arXiv2020

Supervised Discriminative Sparse PCA with Adaptive Neighbors for Dimensionality Reduction

Dimensionality reduction is an important operation in information visualization, feature extraction, clustering, regression, and classification, especially for processing noisy high dimensional data. However, most existing approaches preserve either the global or the local structure of the data, but not both. Approaches that preserve only the global data structure, such as principal component analysis (PCA), are usually sensitive to outliers. Approaches that preserve only the local data structure, such as locality preserving projections, are usually unsupervised (and hence cannot use label information) and uses a fixed similarity graph. We propose a novel linear dimensionality reduction approach, supervised discriminative sparse PCA with adaptive neighbors (SDSPCAAN), to integrate neighborhood-free supervised discriminative sparse PCA and projected clustering with adaptive neighbors. As a result, both global and local data structures, as well as the label information, are used for better dimensionality reduction. Classification experiments on nine high-dimensional datasets validated the effectiveness and robustness of our proposed SDSPCAAN.

cs.LG↗