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Zhenli Xu

Publications and source records attributed to Zhenli Xu.

At least 37 records · Page 2Linked to original sources
arXiv2024

MscaleFNO: Multi-scale Fourier Neural Operator Learning for Oscillatory Function Spaces

In this paper, a multi-scale Fourier neural operator (MscaleFNO) is proposed to reduce the spectral bias of the FNO in learning the mapping between highly oscillatory functions, with application to the nonlinear mapping between the coefficient of the Helmholtz equation and its solution. The MscaleFNO consists of a series of parallel normal FNOs with scaled input of the function and the spatial variable, and their outputs are shown to be able to capture various high-frequency components of the mapping's image. Numerical methods demonstrate the substantial improvement of the MscaleFNO for the problem of wave scattering in the high-frequency regime over the normal FNO with a similar number of network parameters.

math.NA↗
arXiv2024

A fast spectral sum-of-Gaussians method for electrostatic summation in quasi-2D systems

The quasi-2D electrostatic systems, characterized by periodicity in two dimensions with a free third dimension, have garnered significant interest in many fields. We apply the sum-of-Gaussians (SOG) approximation to the Laplace kernel, dividing the interactions into near-field, mid-range, and long-range components. The near-field component, singular but compactly supported in a local domain, is directly calculated. The mid-range component is managed using a procedure similar to nonuniform fast Fourier transforms in three dimensions. The long-range component, which includes Gaussians of large variance, is treated with polynomial interpolation/anterpolation in the free dimension and Fourier spectral solver in the other two dimensions on proxy points. Unlike the fast Ewald summation, which requires extensive zero padding in the case of high aspect ratios, the separability of Gaussians allows us to handle such case without any zero padding in the free direction. Furthermore, while NUFFTs typically rely on certain upsampling in each dimension, and the truncated kernel method introduces an additional factor of upsampling due to kernel oscillation, our scheme eliminates the need for upsampling in any direction due to the smoothness of Gaussians, significantly reducing computational cost for large-scale problems. Finally, whereas all periodic fast multipole methods require dividing the periodic tiling into a smooth far part and a near part containing its nearest neighboring cells, our scheme operates directly on the fundamental cell, resulting in better performance with simpler implementation. We provide a rigorous error analysis showing that upsampling is not required in NUFFT-like steps, achieving $O(N\log N)$ complexity with a small prefactor. The performance of the scheme is demonstrated via extensive numerical experiments.

math.NA↗
arXiv2024

Variance-reduced random batch Langevin dynamics

The random batch method is advantageous in accelerating force calculations in particle simulations, but it poses a challenge of removing the artificial heating effect in application to the Langevin dynamics. We develop an approach to solve this issue by estimating the force variance, resulting in a variance-reduced random batch Langevin dynamics. Theoretical analysis shows the high-order local truncation error of the time step in the numerical discretization scheme, in consistent with the fluctuation-dissipation theorem. Numerical results indicate that the method can achieve a significant variance reduction since a smaller batch size provides accurate approximation, demonstrating the attractive feature of the variance-reduced random batch method for Langevin dynamics.

physics.comp-ph↗
arXiv2024

A divergence-free projection method for quasiperiodic photonic crystals in three dimensions

This paper presents a point-wise divergence-free projection method for numerical approximations of photonic quasicrystals problems. The original three-dimensional quasiperiodic Maxwell's system is transformed into a periodic one in higher dimensions through a variable substitution involving the projection matrix, such that periodic boundary condition can be readily applied. To deal with the intrinsic divergence-free constraint of the Maxwell's equations, we present a quasiperiodic de Rham complex and its associated commuting diagram, based on which a point-wise divergence-free quasiperiodic Fourier spectral basis is proposed. With the help of this basis, we then propose an efficient solution algorithm for the quasiperiodic source problem and conduct its rigorous error estimate. Moreover, by analyzing the decay rate of the Fourier coefficients of the eigenfunctions, we further propose a divergence-free reduced projection method for the quasiperiodic Maxwell eigenvalue problem, which significantly alleviates the computational cost. Several numerical experiments are presented to validate the efficiency and accuracy of the proposed method.

math.NA↗
arXiv2024

RBMD: A molecular dynamics package enabling to simulate 10 million all-atom particles in a single graphics processing unit

This paper introduces a random-batch molecular dynamics (RBMD) package for fast simulations of particle systems at the nano/micro scale. Different from existing packages, the RBMD uses random batch methods for nonbonded interactions of particle systems. The long-range part of Coulomb interactions is calculated in Fourier space by the random batch Ewald algorithm, which achieves linear complexity and superscalability, surpassing classical lattice-based Ewald methods. For the short-range part, the random batch list algorithm is used to construct neighbor lists, significantly reducing both computational and memory costs. The RBMD is implemented on GPU-CPU heterogeneous architectures, with classical force fields for all-atom systems. Benchmark systems are used to validate accuracy and performance of the package. Comparison with the particle-particle particle-mesh method and the Verlet list method in the LAMMPS package is performed on three different NVIDIA GPUs, demonstrating high efficiency of the RBMD on heterogeneous architectures. Our results also show that the RBMD enables simulations on a single GPU with a CPU core up to 10 million particles. Typically, for systems of one million particles, the RBMD allows simulating all-atom systems with a high efficiency of 8.20 ms per step, demonstrating the attractive feature for running large-scale simulations of practical applications on a desktop machine.

physics.comp-ph↗
arXiv2024

Reduced projection method for photonic moiré lattices

This paper presents a reduced projection method for the solution of quasiperiodic Schrödinger eigenvalue problems for photonic moiré lattices. Using the properties of the Schrödinger operator in higher-dimensional space via a projection matrix, we rigorously prove that the generalized Fourier coefficients of the eigenfunctions exhibit faster decay rate along a fixed direction associated with the projection matrix. An efficient reduction strategy of the basis space is then proposed to reduce the degrees of freedom significantly. Rigorous error estimates of the proposed reduced projection method are provided, indicating that a small portion of the degrees of freedom is sufficient to achieve the same level of accuracy as the classical projection method. We present numerical examples of photonic moiré lattices in one and two dimensions to demonstrate the accuracy and efficiency of our proposed method.

math.NA↗
arXiv2024

A note on accurate pressure calculations of Coulomb systems with periodic boundary conditions

In this note, we address some issues concerning the accurate pressure calculation of Coulomb systems with periodic boundary conditions. First, we prove that the formulas for the excess part of the pressure with Ewald summation also reduce to the ensemble average of one-third of the ratio between the potential energy and the volume so that the comments on our previous work in a recent paper by [Onegin~\emph{et al}.,~J. Phys. A: Math.~Theor.~57 (2024) 205002] are incorrect. Second, we demonstrate that in charge non-neutral systems, the pressure expression must be corrected to include interactions with the neutralizing background. This addresses the issues about pressure computation in LAMMPS raised in the paper by Onegin {\it et al.}. Numerical experiments are performed to verify that the pressure obtained via Ewald summation with corrected terms agrees with the average pressure using thermodynamics for the non-neutral OCP system, and are independent of the splitting parameter in the Ewald summation.

physics.comp-ph↗
arXiv2024

Random Batch Ewald Method for Dielectrically Confined Coulomb Systems

Quasi two-dimensional Coulomb systems have drawn widespread interest. The reduced symmetry of these systems leads to complex collective behaviors, yet simultaneously poses significant challenges for particle-based simulations. In this paper, a novel method is presented for efficiently simulate a collection of charges confined in doubly-periodic slabs, with the extension to scenarios involving dielectric jumps at slab boundaries. Unlike existing methods, the method is insensitive to the aspect ratio of simulation box, and it achieves optimal O(N) complexity and strong scalability, thanks to the random batch Ewald (RBE) approach. Moreover, the additional cost for polarization contributions, represented as image reflection series, is reduced to a negligible cost via combining the RBE with an efficient structure factor coefficient re-calibration technique in k-space. Explicit formulas for optimal parameter choices of the algorithm are provided through error estimates, together with a rigorous proof. Finally, we demonstrate the accuracy, efficiency and scalability of our method, called RBE2D, via numerical tests across a variety of prototype systems. An excellent agreement between the RBE2D and the PPPM method is observed, with a significant reduction in the computational cost and strong scalability, demonstrating that it is a promising method for a broad range of charged systems under quasi-2D confinement.

math.NA↗
arXiv2024

TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNs

We introduce the Transformed Generative Pre-Trained Physics-Informed Neural Networks (TGPT-PINN) for accomplishing nonlinear model order reduction (MOR) of transport-dominated partial differential equations in an MOR-integrating PINNs framework. Building on the recent development of the GPT-PINN that is a network-of-networks design achieving snapshot-based model reduction, we design and test a novel paradigm for nonlinear model reduction that can effectively tackle problems with parameter-dependent discontinuities. Through incorporation of a shock-capturing loss function component as well as a parameter-dependent transform layer, the TGPT-PINN overcomes the limitations of linear model reduction in the transport-dominated regime. We demonstrate this new capability for nonlinear model reduction in the PINNs framework by several nontrivial parametric partial differential equations.

math.NA↗
arXiv2023

Energy Stable Scheme for Random Batch Molecular Dynamics

The computational bottleneck of molecular dynamics is the pairwise additive long-range interactions between particles. The random batch Ewald (RBE) method provides a highly efficient and superscalable solver for long-range interactions, but the stochastic nature of this algorithm leads to unphysical self-heating effect during the simulation. We propose an energy stable scheme (ESS) for particle systems by employing a Berendsen-type energy bath. The scheme removes the notorious energy drift which exists due to the force error even when a symplectic integrator is employed. Combining the RBE and the ESS, the new method provides a perfect solution of the computational bottleneck of molecular dynamics at the microcanonical ensemble. Numerical results for primitive electrolyte and all-atom pure water systems demonstrate the attractive performance of the algorithm including its dramatically high accuracy, linear complexity and overcoming the energy drift for long-time simulations.

physics.chem-ph↗
arXiv2023

Local Structure-Preserving Relaxation Method for Charged Systems on Unstructured Meshes

This work considers charged systems described by the modified Poisson--Nernst--Planck (PNP) equations, which incorporate ionic steric effects and the Born solvation energy for dielectric inhomogeneity. Solving the steady-state modified PNP equations poses numerical challenges due to the emergence of sharp boundary layers caused by small Debye lengths, particularly when local ionic concentrations reach saturation. To address this, we first reformulate the steady-state problem as a constraint optimization, where the ionic concentrations on unstructured Delaunay nodes are treated as fractional particles moving along edges between nodes. The electric fields are then updated to minimize the objective free energy while satisfying the discrete Gauss's law. We develop a local relaxation method on unstructured meshes that inherently respects the discrete Gauss's law, ensuring curl-free electric fields. Numerical analysis demonstrates that the optimal mass of the moving fractional particles guarantees the positivity of both ionic and solvent concentrations. Additionally, the free energy of the charged system consistently decreases during successive updates of ionic concentrations and electric fields. We conduct numerical tests to validate the expected numerical accuracy, positivity, free-energy dissipation, and robustness of our method in simulating charged systems with sharp boundary layers.

math.NA↗
arXiv2023

MCMS-RBM: Multi-Component Multi-State Reduced Basis Method toward Efficient Transition Pathway Identification for Crystals and Quasicrystals

Due to quasicrystals having long-range orientational order but without translational symmetry, traditional numerical methods usually suffer when applied as is. In the past decade, the projection method has emerged as a prominent solver for quasiperiodic problems. Transforming them into a higher dimensional but periodic ones, the projection method facilitates the application of the fast Fourier transform. However, the computational complexity inevitably becomes high which significantly impedes e.g. the generation of the phase diagram since a high-fidelity simulation of a problem whose dimension is doubled must be performed for numerous times. To address the computational challenge of quasiperiodic problems based on the projection method, this paper proposes a multi-component multi-state reduced basis method (MCMS-RBM). Featuring multiple components with each providing reduction functionality for one branch of the problem induced by one part of the parameter domain, the MCMS-RBM does not resort to the parameter domain configurations (e.g. phase diagrams) a priori. It enriches each component in a greedy fashion via a phase-transition guided exploration of the multiple states inherent to the problem. Adopting the empirical interpolation method, the resulting online-efficient method vastly accelerates the generation of a delicate phase diagram to a matter of minutes for a parametrized two-turn-four dimensional Lifshitz-Petrich model with two length scales. Moreover, it furnishes surrogate and equally accurate field variables anywhere in the parameter domain.

math.NA↗
arXiv2023

Pythagoras Superposition Principle for Localized Eigenstates of 2D Moiré Lattices

Moiré lattices are aperiodic systems formed by a superposition of two periodic lattices with a relative rotational angle. In optics, the photonic moiré lattice has many appealing properties such as its ability to localize light, thus attracting much attention on exploring features of such a structure. One fundamental research area for photonic moiré lattices is the properties of eigenstates, particularly the existence of localized eigenstates and the localization-to-delocalization transition in the energy band structure. Here we propose an accurate algorithm for the eigenproblems of aperiodic systems by combining plane wave discretization and spectral indicator validation under the higher-dimensional projection, allowing us to explore energy bands of fully aperiodic systems. A localization-delocalization transition regarding the intensity of the aperiodic potential is observed and a novel Pythagoras superposition principle for localized eigenstates of 2D moiré lattices is revealed by analyzing the relationship between the aperiodic and its corresponding periodic eigenstates. This principle sheds light on exploring the physics of localizations for moiré lattice.

physics.optics↗
arXiv2023

How Thermal Effect Regulates Cyclic Voltammetry of Supercapacitors

Cyclic voltammetry (CV) is a powerful technique for characterizing electrochemical properties of electrochemical devices. During charging-discharging cycles, thermal effect has profound impact on its performance, but existing theoretical models cannot clarify such intrinsic mechanism and often give poor prediction. Herein, we propose an interfacial model for the electro-thermal coupling, based on fundamentals in non-equilibrium statistical mechanics. By incorporating molecular interactions, our model shows a quantitative agreement with experimental measurements. The integral capacitance shows a first enhanced then decayed trend against the applied heat bath temperature. Such a relation is attributed to the competition between electrical attraction and Born repulsion via dielectric inhomogeneity, which is rarely understood in previous models. In addition, as evidenced in recent experimental CV tests, our model predicts the non-monotonic dependence of the capacitance on the bulk electrolyte density, further demonstrating its high accuracy. This work demonstrates a potential pathway towards next-generation thermal regulation of electrochemical devices.

cond-mat.mes-hall↗
arXiv2023

Hierarchical Interpolative Factorization for Self Green's Function in 3D Modified Poisson-Boltzmann Equations

The modified Poisson-Boltzmann (MPB) equations are often used to describe equilibrium particle distribution of ionic systems. In this paper, we propose a fast algorithm to solve MPB equations with the self Green's function as the self energy in three dimensions, where the solution of the self Green's function poses a computational bottleneck due to the need to solve a high-dimensional partial differential equation. Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green's function by extending our previous result of two dimensions. This leads to an $O(N\log N)$ algorithm through the strategical utilization of locality and low-rank characteristics of the corresponding operators. Furthermore, the estimated $O(N)$ complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction. Extensive numerical results are performed to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.

physics.comp-ph↗
arXiv2023

Microscopic Energy Storage Mechanism of Dielectric Polymer-Coated Supercapacitors

Supercapacitors have been attracting significant attention as promising energy storage devices. However, the voltage window limitation associated with electrolyte solutions has hindered the improvement of their capacitance. To address this issue and enhance the energy storage capabilities of general traditional supercapacitors, we put forward the dipole induced effects observed in the theoretical framework of the electric double-layer structure. The molecular dynamics results demonstrate that, compared to traditional systems, an improvement of over 50% in integral capacitance at low voltages is achieved. Moreover, a new material-based experimental results obtained from a dielectric supercapacitor employing a hydrated electrolyte solution corroborated the effectiveness of our proposed model, yielding consistent outcomes. We attribute the large capacitance variation to the reorientation of the dipoles, which induces the neutral-to-bilayer transition and the overscreening-to-steric transition, consistent with the polarization process of the polymer in the experiment. We further investigate the capacitance variations under different dipole parameters, such as varying the number of layers, different number densities and different spacings, thereby enriching the experimental results with additional conclusions not previously obtained. This work presents a novel approach that exploits dipole-induced capacitance effects, paving the way for further advances in the field of energy storage technology.

physics.chem-ph↗
arXiv2023

Asymptotic analysis on charging dynamics for stack-electrode model of supercapacitors

Supercapacitors are promising electrochemical energy storage devices due to their prominent performance in rapid charging/discharging rates, long cycle life, stability, etc. Experimental measurement and theoretical prediction on charging timescale for supercapacitors often have large difference. This work develops a matched asymptotic expansion method to derive the charging dynamics of supercapacitors with porous electrodes, in which the supercapacitors are described by the stack-electrode model. Coupling leading-order solutions between every two stacks by continuity of ionic concentration and fluxes leads to an ODE system, which is a generalized equivalent circuit model for zeta potentials, with the potential-dependent nonlinear capacitance and resistance determined by physical parameters of electrolytes, e.g., specific counterion valences for asymmetric electrolytes. Linearized stability analysis on the ODE system after projection is developed to theoretically characterize the charging timescale. The derived asymptotic solutions are numerically verified. Further numerical investigations on the biexponential charging timescales demonstrate that the proposed generalized equivalent circuit model, as well as companion linearized stability analysis, can faithfully capture the charging dynamics of symmetric/asymmetric electrolytes in supercapacitors with porous electrodes.

math.NA↗
arXiv2022

Structure-preserving numerical method for Maxwell-Ampère Nernst-Planck model

Charge dynamics play essential role in many practical applications such as semiconductors, electrochemical devices and transmembrane ion channels. A Maxwell-Ampère Nernst-Planck (MANP) model that describes charge dynamics via concentrations and the electric displacement is able to take effects beyond mean-field approximations into account. To obtain physically faithful numerical solutions, we develop a structure-preserving numerical method for the MANP model whose solution has several physical properties of importance. By the Slotboom transform with entropic-mean approximations, a positivity preserving scheme with Scharfetter-Gummel fluxes is derived for the generalized Nernst-Planck equations. To deal with the curl-free constraint, the dielectric displacement from the Maxwell-Ampère equation is further updated with a local relaxation algorithm of linear computational complexity. We prove that the proposed numerical method unconditionally preserves the mass conservation and the solution positivity at the discrete level, and satisfies the discrete energy dissipation law with a time-step restriction. Numerical experiments verify that our numerical method has expected accuracy and structure-preserving properties. Applications to ion transport with large convection, arising from boundary-layer electric field and Born solvation interactions, further demonstrate that the MANP formulation with the proposed numerical scheme has attractive performance and can effectively describe charge dynamics with large convection of high numerical cell Péclet numbers.

math.NA↗