Jingrun Chen

University of Science and Technology of China | USTC · School of Mathematical Sciences

This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous boundary conditions, including Dirichlet, Neumann, and Robin conditions. We examine MIM with two types of loss functions, referred to as first-order and second-order least squares systems. By providin...

This paper introduces the Asymptotic-Preserving Random Feature Method (APRFM) for the efficient resolution of multiscale radiative transfer equations. The APRFM effectively addresses the challenges posed by stiffness and multiscale characteristics inherent in radiative transfer equations through the application of a micro-macro decomposition strate...

The Landau–Lifshitz–Gilbert (LLG) equation is widely used to model the fast magnetization dynamics of ferromagnets. Recent experimental observations have revealed ultra-fast dynamics at the sub-picosecond timescale, and the inertial LLG equation is proposed to capture the ultra-fast behaviour of magnetization, in which a second temporal derivative...

This paper focuses on solving large-scale, ill-conditioned, and overdetermined sparse least squares problems that arise from numerical partial differential equations (PDEs), mainly from the random feature method. To address these difficulties, we introduce (1) a count sketch technique to sketch the original matrix to a smaller matrix; (2) a QR fact...

DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with discontinuous solutions, DeepONet poses a foundational approximation lower bound due to its linear reconstruction property...

Scientific computing has been an indispensable tool in applied sciences and engineering, where traditional numerical methods are often employed due to their superior accuracy guarantees. However, these methods often encounter challenges when dealing with problems involving complex geometries. Machine learning-based methods, on the other hand, are m...

Machine learning has been widely used for solving partial differential equations (PDEs) in recent years, among which the random feature method (RFM) exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and efficiency. Potentially, the optimization problem in the RFM is more difficult to solve than those that...

Previous works condition aging patterns utilizing one-hot or artificial predefined distributions. Nevertheless, different age groups show different intraclass variations. This property made it challenging to express differences in apparent age across all age groups discriminately. Adaptive aging feature distribution by learning the target age group...

Loss landscape is a useful tool to characterize and compare neural network models. The main challenge for analysis of loss landscape for the deep neural networks is that they are generally highly non-convex in very high dimensional space. In this paper, we develop the "roughness" concept for understanding such landscapes in high dimensions and appl...

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...

Machine learning has been successfully applied to various fields of scientific computing in recent years. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs) with multiscale coefficients. Inspired by the deep mixed residual method, we rewrite the second-order problem...

Interface problems have long been a major focus of scientific computing, leading to the development of various numerical methods. Traditional mesh-based methods often employ time-consuming body-fitted meshes with standard discretization schemes or unfitted meshes with tailored schemes to achieve controllable accuracy and convergence rate. Along ano...

Compared to their three-dimensional counterparts, two-dimensional materials exhibit intriguing electronic and magnetic properties. Notable examples include twisted graphene's superconducting states and chromium trichloride's meron spin textures. Understanding nontrivial topological spin textures is crucial for magnetization dynamics and spintronic...

The diffusion model has shown remarkable success in computer vision, but it remains unclear whether ODE-based probability flow or SDE-based diffusion models are superior and under what circumstances. Comparing the two is challenging due to dependencies on data distribution, score training, and other numerical factors. In this paper, we examine the...

We present a framework for solving time-dependent partial differential equations (PDEs) in the spirit of the random feature method. The numerical solution is constructed using a space-time partition of unity and random feature functions. Two different ways of constructing the random feature functions are investigated: feature functions that treat t...

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)...

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 fi...

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 method...

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...

Microbe-based cancer immunotherapy has recently emerged as a hot topic for cancer treatment. However, serious limitations remain including infection associated side-effect and unsatisfactory outcomes in clinic trials. Here, we fabricate different sizes of nano-formulations derived from yeast cell wall (YCW NPs) by differential centrifugation. The i...

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 residua...

Magnetization dynamics in magnetic materials is often modeled by the Landau-Lifshitz equation, which is solved numerically in general. In micromagnetics simulations, the computational cost relies heavily on the time-marching scheme and the evaluation of the stray field. In this work, we propose a new method, dubbed as GSPM-BDF2, by combining the ad...

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...

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 coefficient m...

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 mag...

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 h...

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, im...

The numerical approximation for the Landau-Lifshitz equation, which models the dynamics of the magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have p...

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 coef...

Solving partial differential equations (PDEs) by parametrizing its solution by neural networks (NNs) has been popular in the past a few years. However, different types of loss functions can be proposed for the same PDE. For the Poisson equation, the loss function can be based on the weak formulation of energy variation or the least squares method,...

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...

A framework for the numerical solution of the Landau-Lifshitz-Gilbert equation is developed in this paper. The numerical framework is based on the finite element method on tetrahedral meshes for the spatial discretization and the implicit midpoint scheme for the temporal discretization. The computational complexity for calculating the demagnetizati...

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)....

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 in...

The semiclassical Schrödinger equation with time-dependent potentials is an important model to study electron dynamics under external controls in the mean-field picture. In this paper, we propose two multiscale finite element methods to solve this problem. In the offline stage, for the first approach, the localized multiscale basis functions are co...

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 networ...

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 residua...

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 str...

Recent years have witnessed growing interests in solving partial differentialequationsbydeepneuralnetworks,especiallyinthehigh-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...

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 sam...

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 sam...

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 classica...

Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on...

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...

A spin wave is the disturbance of intrinsic spin order in magnetic materials. In this paper, a spin wave in the Landau-Lifshitz-Gilbert equation is obtained based on the assumption that the spin wave maintains its shape while it propagates at a constant velocity. Our main findings include: (1) in the absence of Gilbert damping, the spin wave propag...

Micromagnetic simulation is an important tool to study various dynamic behaviors of magnetic order in ferromagnetic materials. The underlying model is the Landau-Lifshitz-Gilbert equation, where the magnetization dynamics is driven by the gyromagnetic torque term and the Gilbert damping term. Numerically, considerable progress has been made in the...

We provide a comprehensive study on the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirichlet boundary condition and transmission condition, subject to the small geometric perturbation and/or the high contrast ratio of the conductivity. All asymptotic terms can be solved in the unpert...

The semiclassical Schr\"odinger equation with time-dependent potentials is an important model to study electron dynamics under external controls in the mean-field picture. In this paper, we propose two multiscale finite element methods to solve this problem. In the offline stage, for the first approach, the localized multiscale basis functions are...

In this paper, we present two improved Gauss-Seidel projection methods with unconditional stability. The first method updates the gyromagnetic term and the damping term simultaneously and follows by a projection step. The second method introduces two sets of approximate solutions, where we update the gyromagnetic term and the damping term simultane...

Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on...

The semiclassical Schr\"{o}dinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wavefunction develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. In this paper, w...

The numerical approximation for the Landau-Lifshitz equation, the dynamics of magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have posed interesting...

In this paper, we use the matched asymptotic analysis to derive a reduced model for domain wall dynamics within thin films of Néel-wall type in soft ferromagnets. It is found that the front dynamics is driven by mean curvature in the absence of external fields, and always relaxes to a planar interface where the one-dimensional ansatz is valid. The...

In recent years, an increasing attention has been paid to quantum heterostructures with tailored functionalities, such as heterojunctions and quantum matematerials, in which quantum dynamics of electrons can be described by the Schr\"{o}dinger equation with multiscale potentials. The model, however, cannot be solved by asymptoics-based approaches w...

Exciton diffusion length plays a vital role in the function of opto-electronic devices. Oftentimes, the domain occupied by an organic semiconductor is subject to surface measurement error. In many experiments, photoluminescence over the domain is measured and used as the observation data to estimate this length parameter in an inverse manner based...

We consider the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirich- let boundary condition and transmission condition, subject to the small geometric perturbation and the high contrast ratio of the conductivity. We consider two types of perturbations: the first corresponds to a thin l...

Static mechanical properties of materials require large-scale nonlinear optimization of the molecular mechanics model under various controls. This paper presents an efficient multigrid strategy to solve such problems. This strategy approximates solutions on grids in a quasi-atomistic and inexact manner, transfers solutions on grids following a coar...

We study the exciton diffusion in organic semiconductors from a macroscopic viewpoint. In a unified way, we conduct the equivalence analysis between Monte-Carlo method and diffusion equation model for photoluminescence quenching and photocurrent spectrum measurements, in both the presence and the absence of Förster energy transfer effect. Connectio...

From a variational perspective, we derive a series of magnetization and quantum spin current systems coupled via an "s-d" potential term, including the Schrödinger-Landau-Lifshitz- Maxwell system, the Pauli-Landau-Lifshitz system, and the Schrödinger-Landau-Lifshitz system with successive simplifications. For the latter two systems, we propose usin...

In this paper, we present a mean-field model of the spin–magnetization coupling in ferromagnetic materials. The model includes non-isotropic diffusion for spin dynamics, which is crucial in capturing strong spin–magnetization coupling. The derivation is based on a moment closure of the quantum spinor dynamics coupled to magnetization dynamics via t...

In this paper, we develop a mean-field model for describing the dynamics of spintransfer torque in multilayered ferromagnetic media. Specifically, we use the techniques of Wigner transform and moment closure to connect the underlying physics at different scales and reach a macroscopic model for the dynamics of spin coupled with the magnetization wi...

In this paper, we propose a wave-equation-based travel-time seismic tomography method with a detailed de-scription of its step-by-step process. First, a linear relation-ship between the travel-time residual = T obs − T syn and the relative velocity perturbation δc(x)/c(x) connected by a finite-frequency travel-time sensitivity kernel K(x) is the-or...

High-resolution 3-D P and S wave crustal velocity and Poisson's ratio models of the 1992 Landers earthquake (Mw 7.3) area are determined iteratively by a wave-equation-based travel-time seismic tomography (WETST) technique. The details of data selection, synthetic arrival-time determination, and trade-off analysis of damping and smoothing parameter...

We study the accuracy of the divide-and-conquer method for electronic structure calculations. The analysis is conducted for a prototypical subdomain problem in the method. We prove that the pointwise difference between electron densities of the global system and the subsystem decays exponentially as a function of the distance away from the boundary...

A common observation from an atomistic to continuum coupling method is that the error is often generated and concentrated near the interface, where the two models are combined. In this paper, a new method is proposed to suppress the error at the interface, and as a consequence, the overall accuracy is improved. The method is motivated by formulatin...