arXiv subjects

Pingwen Zhang

Publications and source records attributed to Pingwen Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2024

Latent assimilation with implicit neural representations for unknown dynamics

Data assimilation is crucial in a wide range of applications, but it often faces challenges such as high computational costs due to data dimensionality and incomplete understanding of underlying mechanisms. To address these challenges, this study presents a novel assimilation framework, termed Latent Assimilation with Implicit Neural Representations (LAINR). By introducing Spherical Implicit Neural Representations (SINR) along with a data-driven uncertainty estimator of the trained neural networks, LAINR enhances efficiency in assimilation process. Experimental results indicate that LAINR holds certain advantage over existing methods based on AutoEncoders, both in terms of accuracy and efficiency.

cs.LG↗

arXiv2024

Theory of polygonal phases self-assembled from T-shaped liquid crystalline molecules

Extensive experimental studies have shown that numerous ordered phases can be formed via the self-assembly of T-shaped liquid crystalline molecules (TLCMs) composed of a rigid backbone, two flexible end chains and a flexible side chain. However, a comprehensive understanding of the stability and formation mechanisms of these intricately nanostructured phases remains incomplete. Here we fill this gap by carrying out a theoretical study of the phase behaviour of TLCMs. Specifically, we construct phase diagrams of TLCMs by computing the free energy of different ordered phases of the system. Our results reveal that the number of polygonal edges increases as the length of side chain or interaction strength increases, consistent with experimental observations. The theoretical study not only reproduces the experimentally observed phases and phase transition sequences, but also systematically analyzes the stability mechanism of the polygonal phases.

cond-mat.soft↗

arXiv2024

Understanding high-index saddle dynamics via numerical analysis

High-index saddle dynamics (HiSD) serves as a competitive instrument in searching the any-index saddle points and constructing the solution landscape of complex systems. The Lagrangian multiplier terms in HiSD ensure the Stiefel manifold constraint, which, however, are dropped in the commonly-used discrete HiSD scheme and are replaced by an additional Gram-Schmidt orthonormalization. Though this scheme has been successfully applied in various fields, it is still unclear why the above modification does not affect its effectiveness. We recover the same form as HiSD from this scheme, which not only leads to error estimates naturally, but indicates that the mechanism of Stiefel manifold preservation by Lagrangian multiplier terms in HiSD is nearly a Gram-Schmidt process (such that the above modification is appropriate). The developed methods are further extended to analyze the more complicated constrained HiSD on high-dimensional sphere, which reveals more mechanisms of the constrained HiSD in preserving several manifold properties.

math.NA↗

arXiv2024

Numerical Methods and Analysis of Computing Quasiperiodic Systems

Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is of great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256: 428, 2014], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudo-spectral) method. Then we analyze the computational complexity of PM and QSM in calculating quasiperiodic systems. The PM can use fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of PM, QSM and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.

math.NA↗

arXiv2024

Accurately recover global quasiperiodic systems by finite points

Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay nor translation invariance. How to accurately recover these systems, especially for non-smooth cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, finite points recovery (FPR) method, which is available for both smooth and non-smooth cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of the quasiperiodic function and the higher-dimensional torus, then recovers the global quasiperiodic system by employing interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of FPR approach in recovering both smooth quasiperiodic functions and piecewise constant Fibonacci quasicrystals. While existing spectral methods encounter difficulties in accurately recovering non-smooth quasiperiodic functions.

math.NA↗

arXiv2023

Semi-implicit method of high-index saddle dynamics and application to construct solution landscape

We analyze the semi-implicit scheme of high-index saddle dynamics, which provides a powerful numerical method for finding the any-index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi-implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi-implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi-implicit scheme via, e.g. technical splittings and multi-variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi-implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi-implicit scheme in constructing solution landscapes of complex systems.

math.NA↗

arXiv2023

Error estimate for semi-implicit method of sphere-constrained high-index saddle dynamics

We prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics, which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems on the high-dimensional sphere. Due to the semi-implicit treatment and the novel computational procedure, the orthonormality of numerical solutions at each time step could not be fully employed to simplify the derivations, and the computations of the state variable and directional vectors are coupled with the retraction, the vector transport and the orthonormalization procedure, which significantly complicates the analysis. We address these issues to prove error estimates for the proposed semi-implicit scheme and then carry out numerical experiments to substantiate the theoretical findings.

math.NA↗

arXiv2022

A model-free shrinking-dimer saddle dynamics for finding saddle point and solution landscape

We propose a model-free shrinking-dimer saddle dynamics for finding any-index saddle points and constructing the solution landscapes, in which the force in the standard saddle dynamics is replaced by a surrogate model trained by the Gassian process learning. By this means, the exact form of the model is no longer necessary such that the saddle dynamics could be implemented based only on some observations of the force. This data-driven approach not only avoids the modeling procedure that could be difficult or inaccurate, but also significantly reduces the number of queries of the force that may be expensive or time-consuming. We accordingly develop a sequential learning saddle dynamics algorithm to perform a sequence of local saddle dynamics, in which the queries of the training samples and the update or retraining of the surrogate force are performed online and around the latent trajectory in order to improve the accuracy of the surrogate model and the value of each sampling. Numerical experiments are performed to demonstrate the effectiveness and efficiency of the proposed algorithm.

math.NA↗

arXiv2022

Error estimates for Euler discretization of high-index saddle dynamics

High-index saddle dynamics provides an effective means to compute the any-index saddle points and construct the solution landscape. In this paper we prove error estimates for Euler discretization of high-index saddle dynamics with respect to the time step size, which remains untreated in the literature. We overcome the main difficulties that lie in the strong nonlinearity of the saddle dynamics and the orthonormalization procedure in the numerical scheme that is uncommon in standard discretization of differential equations. The derived methods are further extended to study the generalized high-index saddle dynamics for non-gradient systems and provide theoretical support for the accuracy of numerical implementations.

math.NA↗

arXiv2022

Mathematical and numerical analysis to shrinking-dimer saddle dynamics with local Lipschitz conditions

We present a mathematical and numerical investigation to the shrinkingdimer saddle dynamics for finding any-index saddle points in the solution landscape. Due to the dimer approximation of Hessian in saddle dynamics, the local Lipschitz assumptions and the strong nonlinearity for the saddle dynamics, it remains challenges for delicate analysis, such as the the boundedness of the solutions and the dimer error. We address these issues to bound the solutions under proper relaxation parameters, based on which we prove the error estimates for numerical discretization to the shrinking-dimer saddle dynamics by matching the dimer length and the time step size. Furthermore, the Richardson extrapolation is employed to obtain a high-order approximation. The inherent reason of requiring the matching of the dimer length and the time step size lies in that the former serves a different mesh size from the later, and thus the proposed numerical method is close to a fully-discrete numerical scheme of some spacetime PDE model with the Hessian in the saddle dynamics and its dimer approximation serving as a "spatial operator" and its discretization, respectively, which in turn indicates the PDE nature of the saddle dynamics.

math.NA↗

arXiv2022

Discretization and index-robust error analysis for constrained high-index saddle dynamics on high-dimensional sphere

We develop and analyze numerical discretization to the constrained high-index saddle dynamics, the dynamics searching for the high-index saddle points confined on the high-dimensional unit sphere. Compared with the saddle dynamics without constraints, the constrained high-index saddle dynamics has more complex dynamical forms, and additional operations such as the retraction and vector transport are required due to the constraint, which significantly complicate the numerical scheme and the corresponding numerical analysis. Furthermore, as the existing numerical analysis results usually depend on the index of the saddle points implicitly, the proved numerical accuracy may be reduced if the index is high in many applications, which indicates the lack of robustness with respect to the index. To address these issues, we derive the error estimates for numerical discretization of the constrained high-index saddle dynamics on high-dimensional sphere, and then improve it by providing an index-robust error analysis in an averaged norm by adjusting the relaxation parameters. The developed results provide mathematical supports for the accuracy of numerical computations.

math.NA↗

arXiv2021

Transition pathways connecting crystals and quasicrystals

Due to structural incommensurability, the emergence of a quasicrystal from a crystalline phase represents a challenge to computational physics. Here the nucleation of quasicrystals is investigated by using an efficient computational method applied to a Landau free-energy functional. Specifically, transition pathways connecting different local minima of the Lifshitz-Petrich model are obtained by using the high-index saddle dynamics. Saddle points on these paths are identified as the critical nuclei of the 6-fold crystals and 12-fold quasicrystals. The results reveal that phase transitions between the crystalline and quasicrystalline phases could follow two possible pathways, corresponding to a one-stage phase transition and a two-stage phase transition involving a metastable lamellar quasicrystalline state, respectively.

cond-mat.mtrl-sci↗

arXiv2021

Solution landscape of the Onsager model identifies non-axisymmetric critical points

We investigate critical points of the Onsager free-energy model on a sphere with different potential kernels, including the dipolar potential, the Maier-Saupe potential, the coupled dipolar/Maier-Saupe potential, and the Onsager potential. A uniform sampling method is implemented for the discretization of the Onsager model, and solution landscapes of the Onsager model are constructed using saddle dynamics coupled with downward/upward search algorithms. We first construct the solution landscapes with the dipolar and Maier-Saupe potentials, for which all critical points are axisymmetric. For the coupled dipolar/Maier-Saupe potential, the solution landscape shows a novel non-axisymmetric critical point, named tennis, which exists for a wide range of parameters. We further demonstrate various non-axisymmetric critical points in the Onsager model with the Onsager potential, including square, hexagon, octahedral, cubic, quarter, icosahedral}, and dodecahedral states. The bifurcation diagram is presented to show the primary and secondary bifurcations of the isotropic state and reveal the emergence of the critical points. The solution landscape provides an efficient approach to show the global structure of the model system as well as the bifurcations of critical points, which can not only support the previous theoretical conjectures but also propose new conjectures based on the numerical findings.

math.NA↗

arXiv2021

Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.

math.NA↗

arXiv2021

Construction of a pathway map on a complicated energy landscape

How do we search for the entire family tree without unwanted random guesses, starting from a high-index and high-energy stationary state on the energy landscape? Here we introduce a general numerical method that constructs the pathway map clearly connecting all stationary points branched from a single parent state. The map guides our understanding of how a physical system moves on the energy landscape. In particular, the method allows us to identify the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landau-de Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.

physics.comp-ph↗

arXiv2021

Modeling and Computation of Liquid Crystals

Liquid crystal is a typical kind of soft matter that is intermediate between crystalline solids and isotropic fluids. The study of liquid crystals has made tremendous progress over the last four decades, which is of great importance on both fundamental scientific researches and widespread applications in industry. In this paper, we review the mathematical models and their connections of liquid crystals, and survey the developments of numerical methods for finding the rich configurations of liquid crystals.

math.NA↗

arXiv2020

Solution landscape of a reduced Landau-de Gennes model on a hexagon

We investigate the solution landscape of a reduced Landau--de Gennes model for nematic liquid crystals on a two-dimensional hexagon at a fixed temperature, as a function of $λ$---the edge length. This is a generic example for reduced approaches on regular polygons. We apply the high-index optimization-based shrinking dimer method to systematically construct the solution landscape consisting of multiple defect solutions and relationships between them. We report a new stable T state with index-$0$ that has an interior $-1/2$ defect; new classes of high-index saddle points with multiple interior defects referred to as H class and TD class; changes in the Morse index of saddle points with $λ^2$ and novel pathways mediated by high-index saddle points that can control and steer dynamical pathways. The range of topological degrees, locations and multiplicity of defects offered by these saddle points can be used to navigate through complex solution landscapes of nematic liquid crystals and other related soft matter systems.

math-ph↗

arXiv2020

Detecting Suspected Epidemic Cases Using Trajectory Big Data

Emerging infectious diseases are existential threats to human health and global stability. The recent outbreaks of the novel coronavirus COVID-19 have rapidly formed a global pandemic, causing hundreds of thousands of infections and huge economic loss. The WHO declares that more precise measures to track, detect and isolate infected people are among the most effective means to quickly contain the outbreak. Based on trajectory provided by the big data and the mean field theory, we establish an aggregated risk mean field that contains information of all risk-spreading particles by proposing a spatio-temporal model named HiRES risk map. It has dynamic fine spatial resolution and high computation efficiency enabling fast update. We then propose an objective individual epidemic risk scoring model named HiRES-p based on HiRES risk maps, and use it to develop statistical inference and machine learning methods for detecting suspected epidemic-infected individuals. We conduct numerical experiments by applying the proposed methods to study the early outbreak of COVID-19 in China. Results show that the HiRES risk map has strong ability in capturing global trend and local variability of the epidemic risk, thus can be applied to monitor epidemic risk at country, province, city and community levels, as well as at specific high-risk locations such as hospital and station. HiRES-p score seems to be an effective measurement of personal epidemic risk. The accuracy of both detecting methods are above 90\% when the population infection rate is under 20\%, which indicates great application potential in epidemic risk prevention and control practice.

stat.AP↗