arXiv subjects

Jingrun Chen

Publications and source records attributed to Jingrun Chen.

At least 55 records · Page 3Linked to original sources

arXiv2023

Micromagnetics simulations and phase transitions of ferromagnetics with Dzyaloshinskii-Moriya interaction

Magnetic skyrmions widely exist in a diverse range of magnetic systems, including chiral magnets with a non-centrosymmetric structure characterized by Dzyaloshinkii-Moriya interaction~(DMI). In this study, we propose a generalized semi-implicit backward differentiation formula projection method, enabling the simulations of the Landau-Lifshitz~(LL) equation in chiral magnets in a typical time step-size of $1$ ps, markedly exceeding the limit subjected by existing numerical methods of typically $0.1$ ps. Using micromagnetics simulations, we show that the LL equation with DMI reveals an intriguing dynamic instability in magnetization configurations as the damping varies. Both the isolated skyrmionium and skyrmionium clusters can be consequently produced using a simple initialization strategy and a specific damping parameter. Assisted by the string method, the transition path between skyrmion and skyrmionium, along with the escape of a skyrmion from the skyrmion clusters, are then thoroughly examined. The numerical methods developed in this work not only provide a reliable paradigm to investigate the skyrmion-based textures and their transition paths, but also facilitate the understandings for magnetization dynamics in complex magnetic systems.

math.NA↗

arXiv2022

Convergence analysis of an implicit finite difference method for the inertial Landau-Lifshitz-Gilbert equation

The Landau-Lifshitz-Gilbert (LLG) equation is a widely used model for fast magnetization dynamics in ferromagnetic materials. Recently, the inertial LLG equation, which contains an inertial term, has been proposed to capture the ultra-fast magnetization dynamics at the sub-picosecond timescale. Mathematically, this generalized model contains the first temporal derivative and a newly introduced second temporal derivative of magnetization. Consequently, it produces extra difficulties in numerical analysis due to the mixed hyperbolic-parabolic type of this equation with degeneracy. In this work, we propose an implicit finite difference scheme based on the central difference in both time and space. A fixed point iteration method is applied to solve the implicit nonlinear system. With the help of a second order accurate constructed solution, we provide a convergence analysis in $H^1$ for this numerical scheme, in the $\ell^\infty (0, T; H_h^1)$ norm. It is shown that the proposed method is second order accurate in both time and space, with unconditional stability and a natural preservation of the magnetization length. In the hyperbolic regime, significant damping wave behaviors of magnetization at a shorter timescale are observed through numerical simulations.

math.NA↗

arXiv2022

Bridging Traditional and Machine Learning-based Algorithms for Solving PDEs: The Random Feature Method

One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A long list of numerical methods have been proposed and successfully used for various applications. In recent years, deep learning methods have shown their superiority for high-dimensional PDEs where traditional methods fail. However, for low dimensional problems, it remains unclear whether these methods have a real advantage over traditional algorithms as a direct solver. In this work, we propose the random feature method (RFM) for solving PDEs, a natural bridge between traditional and machine learning-based algorithms. RFM is based on a combination of well-known ideas: 1. representation of the approximate solution using random feature functions; 2. collocation method to take care of the PDE; 3. the penalty method to treat the boundary conditions, which allows us to treat the boundary condition and the PDE in the same footing. We find it crucial to add several additional components including multi-scale representation and rescaling the weights in the loss function. We demonstrate that the method exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and efficiency. In addition, we find that RFM is particularly suited for complex problems with complex geometry, where both traditional and machine learning-based algorithms encounter difficulties.

math.NA↗

arXiv2022

Homogenization of the Landau-Lifshitz-Gilbert equation with natural boundary condition

The full Landau-Lifshitz-Gilbert equation with periodic material coefficients and natural boundary condition is employed to model the magnetization dynamics in composite ferromagnets. In this work, we establish the convergence between the homogenized solution and the original solution via a Lax equivalence theorem kind of argument. There are a few technical difficulties, including: 1) it is proven the classic choice of corrector to homogenization cannot provide the convergence result in the $H^1$ norm; 2) a boundary layer is induced due to the natural boundary condition; 3) the presence of stray field give rise to a multiscale potential problem. To keep the convergence rates near the boundary, we introduce the Neumann corrector with a high-order modification. Estimates on singular integral for disturbed functions and boundary layer are deduced, to conduct consistency analysis of stray field. Furthermore, inspired by length conservation of magnetization, we choose proper correctors in specific geometric space. These, together with a uniform $W^{1,6}$ estimate on original solution, provide the convergence rates in the $H^1$ sense.

math.AP↗

arXiv2021

Convergence Analysis of A Second-order Accurate, Linear Numerical Scheme for The Landau-Lifshitz Equation with Large Damping Parameters

A second order accurate, linear numerical method is analyzed for the Landau-Lifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a non-convexity constraint of unit length of the magnetization. The numerical method is based on the second-order backward differentiation formula in time, combined with an implicit treatment of the linear diffusion term and explicit extrapolation for the nonlinear terms. Afterward, a projection step is applied to normalize the numerical solution at a point-wise level. This numerical scheme has shown extensive advantages in the practical computations for the physical model with large damping parameters, which comes from the fact that only a linear system with constant coefficients (independent of both time and the updated magnetization) needs to be solved at each time step, and has greatly improved the numerical efficiency. Meanwhile, a theoretical analysis for this linear numerical scheme has not been available. In this paper, we provide a rigorous error estimate of the numerical scheme, in the discrete $\ell^{\infty}(0,T; \ell^2) \cap \ell^2(0,T; H_h^1)$ norm, under suitable regularity assumptions and reasonable ratio between the time step-size and the spatial mesh-size. In particular, the projection operation is nonlinear, and a stability estimate for the projection step turns out to be highly challenging. Such a stability estimate is derived in details, which will play an essential role in the convergence analysis for the numerical scheme, if the damping parameter is greater than 3.

math.NA↗

arXiv2021

A second-order semi-implicit method for the inertial Landau-Lifshitz-Gilbert equation

Recent theoretical and experimental advances show that the inertia of magnetization emerges at sub-picoseconds and contributes to the ultrafast magnetization dynamics which cannot be captured intrinsically by the LLG equation. Therefore, as a generalization, the inertial Landau-Lifshitz-Gilbert (iLLG) equation is proposed to model the ultrafast magnetization dynamics. Mathematically, the LLG equation is a nonlinear system of parabolic type with (possible) degeneracy. However, the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy, and exhibits more complicated structures. It behaves like a hyperbolic system at the sub-picosecond scale while behaves like a parabolic system at larger timescales. Such hybrid behaviors impose additional difficulties on designing numerical methods for the iLLG equation. In this work, we propose a second-order semi-implicit scheme to solve the iLLG equation. The second temporal derivative of magnetization is approximated by the standard centered difference scheme and the first derivative is approximated by the midpoint scheme involving three time steps. The nonlinear terms are treated semi-implicitly using one-sided interpolation with the second-order accuracy. At each step, the unconditionally unique solvability of the unsymmetric linear system of equations in the proposed method is proved with a detailed discussion on the condition number. Numerically, the second-order accuracy in both time and space is verified. Using the proposed method, the inertial effect of ferromagnetics is observed in micromagnetics simulations at small timescales, in consistency with the hyperbolic property of the model at sub-picoseconds. For long time simulations, the results of the iLLG model are in nice agreements with those of the LLG model, in consistency with the parabolic feature of the iLLG model at larger timescales.

math.NA↗

arXiv2021

A Machine-Learning Method for Time-Dependent Wave Equations over Unbounded Domains

Time-dependent wave equations represent an important class of partial differential equations (PDE) for describing wave propagation phenomena, which are often formulated over unbounded domains. Given a compactly supported initial condition, classical numerical methods reduce such problems to bounded domains using artificial boundary condition (ABC). In this work, we present a machine-learning method to solve this type of equations as an alternative to ABCs. Specifically, the mapping from the initial conditions to the PDE solution is represented by a neural network, trained using wave packets that are parameterized by their band width and wave numbers. The accuracy is tested for both the second-order wave equation and the Schrodinger equation, including the nonlinear Schrodinger equation. We examine the accuracy from both interpolations and extrapolations. For initial conditions lying in the training set, the learned map has good interpolation accuracy, due to the approximation property of deep neural networks. The learned map also exhibits some good extrapolation accuracy. We also demonstrate the effectiveness of the method for problems in irregular domains. Overall, the proposed method provides an interesting alternative for finite-time simulation of wave propagation.

math.NA↗

arXiv2021

A Deep Learning Based Discontinuous Galerkin Method for Hyperbolic Equations with Discontinuous Solutions and Random Uncertainties

We propose a deep learning based discontinuous Galerkin method (D2GM) to solve hyperbolic equations with discontinuous solutions and random uncertainties. The main computational challenges for such problems include discontinuities of the solutions and the curse of dimensionality due to uncertainties. Deep learning techniques have been favored for high-dimensional problems but face difficulties when the solution is not smooth, thus have so far been mainly used for viscous hyperbolic system that admits only smooth solutions. We alleviate this difficulty by setting up the loss function using discrete shock capturing schemes--the discontinous Galerkin method as an example--since the solutions are smooth in the discrete space. The convergence of D2GM is established via the Lax equivalence theorem kind of argument. The high-dimensional random space is handled by the Monte-Carlo method. Such a setup makes the D2GM approximate high-dimensional functions over the random space with satisfactory accuracy at reasonable cost. The D2GM is found numerically to be first-order and second-order accurate for (stochastic) linear conservation law with smooth solutions using piecewise constant and piecewise linear basis functions, respectively. Numerous examples are given to verify the efficiency and the robustness of D2GM with the dimensionality of random variables up to $200$ for (stochastic) linear conservation law and (stochastic) Burgers' equation.

math.NA↗

arXiv2021

Advantages of a semi-implicit scheme over a fully implicit scheme for Landau-Lifshitz-Gilbert equation

Magnetization dynamics in magnetic materials is modeled by the Landau-Lifshitz-Gilbert (LLG) equation. In the LLG equation, the length of magnetization is conserved and the system energy is dissipative. Implicit and semi-implicit schemes have been used in micromagnetics simulations due to their unconditional numerical stability. In more details, implicit schemes preserve the properties of the LLG equation, but solve a nonlinear system of equations per time step. In contrast, semi-implicit schemes only solve a linear system of equations, while additional operations are needed to preserve the length of magnetization. It still remains unclear which one shall be used if both implicit and semi-implicit schemes are available. In this work, using the implicit Crank-Nicolson (ICN) scheme as a benchmark, we propose to make this implicit scheme semi-implicit. It can be proved that both schemes are second-order accurate in space and time. For the unique solvability of nonlinear systems of equations in the ICN scheme, we require that the temporal step size scales quadratically with the spatial mesh size. It is numerically verified that the convergence of the nonlinear solver becomes slower for larger temporal step size and multiple magnetization profiles are obtained for different initial guesses. The linear systems of equations in the semi-implicit CN (SICN) scheme are unconditionally uniquely solvable, and the condition that the temporal step size scales linearly with the spatial mesh size is needed in the convergence of the SICN scheme. In terms of numerical efficiency, the SICN scheme achieves the same accuracy as the ICN scheme with less computational time. Based on these results, we conclude that a semi-implicit scheme is superior to its implicit analog both theoretically and numerically, and we recommend the semi-implicit scheme in micromagnetics simulations if both methods are available.

physics.comp-ph↗

arXiv2021

A second-order numerical method for Landau-Lifshitz-Gilbert equation with large damping parameters

A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method are associated with the following features: (1) It only solves linear systems of equations with constant coefficients where fast solvers are available, so that the numerical efficiency has been greatly improved, in comparison with the existing Gauss-Seidel project method. (2) The second-order accuracy in time is achieved, and it is unconditionally stable for large damping parameters. Moreover, both the second-order accuracy and the great efficiency improvement will be verified by several numerical examples in the 1D and 3D simulations. In the presence of large damping parameters, it is observed that this method is unconditionally stable and finds physically reasonable structures while many existing methods have failed. For the domain wall dynamics, the linear dependence of wall velocity with respect to the damping parameter and the external magnetic field will be obtained through the reported simulations.

physics.comp-ph↗

arXiv2021

A Gauss-Seidel projection method with the minimal number of updates for stray field in micromagnetic simulations

Magnetization dynamics in magnetic materials is often modeled by the Landau-Lifshitz equation, which is solved numerically in general. In micromagnetic simulations, the computational cost relies heavily on the time-marching scheme and the evaluation of stray field. Explicit marching schemes are efficient but suffer from severe stability constraints, while nonlinear systems of equations have to be solved in implicit schemes though they are unconditionally stable. A better compromise between stability and efficiency is the semi-implicit scheme, such as the Gauss-Seidel projection method (GSPM) and the second-order backward differentiation formula scheme (BDF2). At each marching step, GSPM solves several linear systems of equations with constant coefficients and updates the stray field several times, while BDF2 updates the stray field only once but solves a larger linear system of equations with variable coefficients and a nonsymmetric structure. In this work, we propose a new method, dubbed as GSPM-BDF2, by combing the advantages of both GSPM and BDF2. Like GSPM, this method is first-order accurate in time and second-order accurate in space, and is unconditionally stable with respect to the damping parameter. However, GSPM-BDF2 updates the stray field only once per time step, leading to an efficiency improvement of about $60\%$ than the state-of-the-art GSPM for micromagnetic simulations. For Standard Problem \#4 and \#5 from National Institute of Standards and Technology, GSPM-BDF2 reduces the computational time over the popular software OOMMF by $82\%$ and $96\%$, respectively. Thus, the proposed method provides a more efficient choice for micromagnetic simulations.

math.NA↗

arXiv2020

A consensus-based global optimization method with adaptive momentum estimation

Objective functions in large-scale machine-learning and artificial intelligence applications often live in high dimensions with strong non-convexity and massive local minima. First-order methods, such as the stochastic gradient method and Adam, are often used to find global minima. Recently, the consensus-based optimization (CBO) method has been introduced as one of the gradient-free optimization methods and its convergence is proven with dimension-dependent parameters, which may suffer from the curse of dimensionality. By replacing the isotropic geometric Brownian motion with the component-wise one, the latest improvement of the CBO method is guaranteed to converge to the global minimizer with dimension-independent parameters, although the initial data need to be well-chosen. In this paper, based on the CBO method and Adam, we propose a consensus-based global optimization method with adaptive momentum estimation (Adam-CBO). Advantages of the Adam-CBO method include: (1) capable of finding global minima of non-convex objective functions with high success rates and low costs; (2) can handle non-differentiable activation functions and thus approximate low-regularity functions with better accuracy. The former is verified by approximating the $1000$ dimensional Rastrigin function with $100\%$ success rate at a cost only growing linearly with respect to the dimensionality. The latter is confirmed by solving a machine learning task for partial differential equations with low-regularity solutions where the Adam-CBO method provides better results than the state-of-the-art method Adam. A linear stability analysis is provided to understand the asymptotic behavior of the Adam-CBO method.

math.OC↗

arXiv2020

A QMC-deep learning method for diffusivity estimation in random domains

Exciton diffusion plays a vital role in the function of many organic semiconducting opto-electronic devices, where an accurate description requires precise control of heterojunctions. This poses a challenging problem because the parameterization of heterojunctions in high-dimensional random space is far beyond the capability of classical simulation tools. Here, we develop a novel method based on quasi-Monte Carlo sampling to generate the training data set and deep neural network to extract a function for exciton diffusion length on surface roughness with high accuracy and unprecedented efficiency, yielding an abundance of information over the entire parameter space. Our method provides a new strategy to analyze the impact of interfacial ordering on exciton diffusion and is expected to assist experimental design with tailored opto-electronic functionalities.

physics.comp-ph↗

arXiv2020

Enforcing exact boundary and initial conditions in the deep mixed residual method

In theory, boundary and initial conditions are important for the wellposedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method and finite element method. Recent years have witnessed growing interests in solving PDEs by deep neural networks (DNNs), especially in the high-dimensional case. However, in the generic situation, a careful literature review shows that boundary conditions cannot be enforced exactly for DNNs, which inevitably leads to a modeling error. In this work, based on the recently developed deep mixed residual method (MIM), we demonstrate how to make DNNs satisfy boundary and initial conditions automatically in a systematic manner. As a consequence, the loss function in MIM is free of the penalty term and does not have any modeling error. Using numerous examples, including Dirichlet, Neumann, mixed, Robin, and periodic boundary conditions for elliptic equations, and initial conditions for parabolic and hyperbolic equations, we show that enforcing exact boundary and initial conditions not only provides a better approximate solution but also facilitates the training process.

math.NA↗

arXiv2020

A comparison study of deep Galerkin method and deep Ritz method for elliptic problems with different boundary conditions

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty method. In the work, we conduct a comparison study for elliptic problems with four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin method and deep Ritz method. In the former, the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions. Besides, in some cases, when the boundary condition can be implemented in an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.

math.NA↗

arXiv2020

MIM: A deep mixed residual method for solving high-order partial differential equations

In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the least-squares sense as the loss function and a deep neural network (DNN) to approximate the PDE solution. In this work, we propose a deep mixed residual method (MIM) to solve PDEs with high-order derivatives. In MIM, we first rewrite a high-order PDE into a first-order system, very much in the same spirit as local discontinuous Galerkin method and mixed finite element method in classical numerical methods for PDEs. We then use the residual of first-order system in the least-squares sense as the loss function, which is in close connection with least-squares finite element method. For aforementioned classical numerical methods, the choice of trail and test functions is important for stability and accuracy issues in many cases. MIM shares this property when DNNs are employed to approximate unknowns functions in the first-order system. In one case, we use nearly the same DNN to approximate all unknown functions and in the other case, we use totally different DNNs for different unknown functions. In most cases, MIM provides better approximations (not only for high-derivatives of the PDE solution but also for the PDE solution itself) than DGM with nearly the same DNN and the same execution time, sometimes by more than one order of magnitude. When different DNNs are used, in many cases, MIM provides even better approximations than MIM with only one DNN, sometimes by more than one order of magnitude. Therefore, we expect MIM to open up a possibly systematic way to understand and improve deep learning for solving PDEs from the perspective of classical numerical analysis.

math.NA↗

arXiv2019

Quasi-Monte Carlo sampling for machine-learning partial differential equations

Solving partial differential equations in high dimensions by deep neural network has brought significant attentions in recent years. In many scenarios, the loss function is defined as an integral over a high-dimensional domain. Monte-Carlo method, together with the deep neural network, is used to overcome the curse of dimensionality, while classical methods fail. Often, a deep neural network outperforms classical numerical methods in terms of both accuracy and efficiency. In this paper, we propose to use quasi-Monte Carlo sampling, instead of Monte-Carlo method to approximate the loss function. To demonstrate the idea, we conduct numerical experiments in the framework of deep Ritz method proposed by Weinan E and Bing Yu. For the same accuracy requirement, it is observed that quasi-Monte Carlo sampling reduces the size of training data set by more than two orders of magnitude compared to that of MC method. Under some assumptions, we prove that quasi-Monte Carlo sampling together with the deep neural network generates a convergent series with rate proportional to the approximation accuracy of quasi-Monte Carlo method for numerical integration. Numerically the fitted convergence rate is a bit smaller, but the proposed approach always outperforms Monte Carlo method. It is worth mentioning that the convergence analysis is generic whenever a loss function is approximated by the quasi-Monte Carlo method, although observations here are based on deep Ritz method.

math.NA↗

arXiv2019

Numerical methods for antiferromagnetics

Compared with ferromagnetic counterparts, antiferromagnetic materials are considered as the future of spintronic applications since these materials are robust against the magnetic perturbation, produce no stray field, and display ultrafast dynamics. There are (at least) two sets of magnetic moments in antiferromagnets (with magnetization of the same magnitude but antiparallel directions) and ferrimagnets (with magnetization of the different magnitude). The coupled dynamics for the bipartite collinear antiferromagnets is modeled by a coupled system of Landau-Lifshitz-Gilbert equations with an additional term originated from the antiferromagnetic exchange, which leads to femtosecond magnetization dynamics. In this paper, we develop three Gauss-Seidel projection methods for micromagnetics simulation in antiferromagnets and ferrimagnets. They are first-order accurate in time and second-order in space, and only solve linear systems of equations with constant coefficients at each step. Femtosecond dynamics, Néel wall structure, and phase transition in presence of an external magnetic field for antiferromagnets are provided with the femtosecond stepsize.

cs.CE↗