In this paper, we present linearized learning methods to accelerate the convergence of training for stationary nonlinear Navier-Stokes equations. To solve the stationary nonlinear Navier-Stokes (NS) equation, we integrate the procedure of linearization of the nonlinear convection term in the NS equation into the training process of multi-scale deep neural network approximation of the NS solution. Four forms of linearizations are considered. After a benchmark problem, we solve the highly oscillating stationary flows utilizing the proposed linearized learning with multi-scale neural network for complex domains. The results show that multiscale deep neural network combining with the linearized schemes can be trained fast and accurately.
In this paper, we propose a multiscale DeepONet to represent nonlinear operator between Banach spaces of highly oscillatory continuous functions. The multiscale deep neural network (DNN) utilizes a multiple scaling technique to convert high frequency function to lower frequency functions before using a DNN to learn a specific range of frequency of the function. The multi-scale concept is integrated into the DeepONet which is based on a universal approximation theory of nonlinear operators. The resulting multi-scale DeepONet is shown to be effective to represent building seismic response operat…
In this paper, we consider several possible ways to set up Heterogeneous Multiscale Methods for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient, which can be seen as a means to modeling rapidly varying ferromagnetic materials. We then prove estimates for the errors introduced when approximating the relevant quantity in each of the models given a periodic problem, using averaging in time and space of the solution to a corresponding micro problem. In our setup, the Landau-Lifshitz equation with highly oscillatory coefficient is chosen as the micro problem for all mod…
Nonlinear Geometric Optics Based Multiscale Stochastic Galerkin Methods for Highly Oscillatory Transport Equations with Random Inputs
Inria (IPSO team), IRMAR, University of Rennes 1（法国国家信息与自动化研究所（IPSO团队），雷恩第一大学）；Department of Mathematics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校数学系）；CNRS, IRMAR, Inria (IPSO team), University of Rennes 1（法国国家科学研究中心，法国国家信息与自动化研究所（IPSO团队），雷恩第一大学）
Parareal methods for highly oscillatory dynamical systems
Bar-Ilan University（巴伊兰大学）；Georgia Institute of Technology（佐治亚理工学院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；KTH Royal Institute of Technology（瑞典皇家理工学院）
In this paper, we present a finite difference heterogeneous multiscale method for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient. The approach combines a higher order discretization and artificial damping in the so-called micro problem to obtain an efficient implementation. The influence of different parameters on the resulting approximation error is discussed. Numerical examples for both periodic as well as more general coefficients are given to demonstrate the functionality of the approach.
High-order stroboscopic averaging methods for highly oscillatory delay problems
Universidad de Valladolid（巴利亚多利德大学）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学与应用科学中心）
We present a Heterogeneous Multiscale Method for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient, a simple model for a ferromagnetic composite. A finite element macro scheme is combined with a finite difference micro model to approximate the effective equation corresponding to the original problem. This makes it possible to obtain effective solutions to problems with rapid material variations on a small scale, described by $\varepsilon \ll 1$, which would be too expensive to resolve in a conventional simulation.
High order multiscale methods for advection-diffusion equation in highly oscillatory regimes: application to surfactant diffusion and generalization to arbitrary domains
The flow regime of micro flow varies from collisionless regime to hydrodynamic regime according to the Knudsen number. On the kinetic scale, the dynamics of micro flow can be described by the linearized kinetic equation. In the continuum regime, hydrodynamic equations such as linearized Navier-Stokes equations and Euler equations can be derived from the linearized kinetic equation by the Chapman-Enskog asymptotic analysis. In this paper, based on the linearized kinetic equation we are going to propose a unified gas kinetic scheme scheme (UGKS) for micro flow simulation, which is an effective m…
In fluid flow simulation, the multi-continuum model is a useful strategy. When the heterogeneity and contrast of coefficients are high, the system becomes multiscale, and some kinds of reduced-order methods are demanded. Combining these techniques with nonlinearity, we will consider in this paper a dual-continuum model which is generalized as a multi-continuum model for a coupled system of nonlinear Richards equations as unsaturated flows, in complex heterogeneous fractured porous media; and we will solve it by a novel multiscale approach utilizing the constraint energy minimizing generalized …
We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with $H^1$-conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocit…
We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress local solution manifolds. Our framework is applied to a semilinear elliptic equation with oscillatory media and a nonlinear radiative transfer equation; in both cases, significant improvements in efficacy are observed. This new method does not rely on detailed analytical understanding of the multiscale PDEs, such as their asymptotic limits, and thus is more vers…
In this paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to first solving a nonlinear poroelasticity problem. The arising system consists of a nonlinear pressure equation and a nonlinear stress equation in strain-limiting setting, where strains keep bounded while stresses can grow arbitrarily large. After time discretization of the system, to tackle the nonlinearity, we linearize the resulting equations by Picard iteration. To handle the linearized equations, we employ the CEM-GMsFEM and obtain appropriate offline multiscale basis funct…
We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an ${LU}$-like hierarchical factorization of the dense system matrix, permitting application of the inverse in $\mathcal O(n)$ time, where $n$ is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted adapt…
This paper is concerned with a blood flow problem coupled with a slow plaque growth at the artery wall. In the model, the micro (fast) system is the Navier-Stokes equation with a periodically applied force and the macro (slow) system is a fractional reaction equation, which is used to describe the plaque growth with memory effect. We construct an auxiliary temporal periodic problem and an effective time-average equation to approximate the original problem and analyze the approximation error of the corresponding linearized PDE (Stokes) system, where the simple front-tracking technique is used t…
Beijing Normal-Hong Kong Baptist University（北京岭南师范学院）；Guangzhou Nanfang College（广州南方学院）；Louisiana State University（路易斯安那州立大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime
National University of Singapore（新加坡国立大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Purdue University（普渡大学）；Université de Lorraine（洛林大学）；Inria Nancy-Grand Est（南锡大东部国家研究学院）
A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs), which are related to the FBSDEs by the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the pathwise difference between two discrete stochastic processes, defined by the time discretization of the FBSDEs and the DNN representation of the PDE solutions, respectively. The proposed algorithms are shown to generate DNN solutions for a…
Uniformly accurate exponential-type integrators for klein-gordon equations with asymptotic convergence to classical splitting schemes in the nonlinear schroedinger limit
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；INRIA & ENS Rennes（法国国家信息与自动化研究所与雷恩高等师范学院）
