33 research outputs found

    Hierarchical deep learning-based adaptive time-stepping scheme for multiscale simulations

    Full text link
    Multiscale is a hallmark feature of complex nonlinear systems. While the simulation using the classical numerical methods is restricted by the local \textit{Taylor} series constraints, the multiscale techniques are often limited by finding heuristic closures. This study proposes a new method for simulating multiscale problems using deep neural networks. By leveraging the hierarchical learning of neural network time steppers, the method adapts time steps to approximate dynamical system flow maps across timescales. This approach achieves state-of-the-art performance in less computational time compared to fixed-step neural network solvers. The proposed method is demonstrated on several nonlinear dynamical systems, and source codes are provided for implementation. This method has the potential to benefit multiscale analysis of complex systems and encourage further investigation in this area

    Multi-Scale Simulation of Complex Systems: A Perspective of Integrating Knowledge and Data

    Full text link
    Complex system simulation has been playing an irreplaceable role in understanding, predicting, and controlling diverse complex systems. In the past few decades, the multi-scale simulation technique has drawn increasing attention for its remarkable ability to overcome the challenges of complex system simulation with unknown mechanisms and expensive computational costs. In this survey, we will systematically review the literature on multi-scale simulation of complex systems from the perspective of knowledge and data. Firstly, we will present background knowledge about simulating complex system simulation and the scales in complex systems. Then, we divide the main objectives of multi-scale modeling and simulation into five categories by considering scenarios with clear scale and scenarios with unclear scale, respectively. After summarizing the general methods for multi-scale simulation based on the clues of knowledge and data, we introduce the adopted methods to achieve different objectives. Finally, we introduce the applications of multi-scale simulation in typical matter systems and social systems

    NySALT: Nystr\"{o}m-type inference-based schemes adaptive to large time-stepping

    Full text link
    Large time-stepping is important for efficient long-time simulations of deterministic and stochastic Hamiltonian dynamical systems. Conventional structure-preserving integrators, while being successful for generic systems, have limited tolerance to time step size due to stability and accuracy constraints. We propose to use data to innovate classical integrators so that they can be adaptive to large time-stepping and are tailored to each specific system. In particular, we introduce NySALT, Nystr\"{o}m-type inference-based schemes adaptive to large time-stepping. The NySALT has optimal parameters for each time step learnt from data by minimizing the one-step prediction error. Thus, it is tailored for each time step size and the specific system to achieve optimal performance and tolerate large time-stepping in an adaptive fashion. We prove and numerically verify the convergence of the estimators as data size increases. Furthermore, analysis and numerical tests on the deterministic and stochastic Fermi-Pasta-Ulam (FPU) models show that NySALT enlarges the maximal admissible step size of linear stability, and quadruples the time step size of the St\"{o}rmer--Verlet and the BAOAB when maintaining similar levels of accuracy.Comment: 26 pages, 7 figure

    A symmetry-adapted numerical scheme for SDEs

    Get PDF
    We propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry adapted coordinates system where the given SDE has notable invariant properties. An approximation scheme preserving the symmetry properties of the equation is introduced. Our algorithmic procedure is applied to the family of general linear SDEs for which two theoretical estimates of the numerical forward error are established.Comment: A numerical example adde

    A class of robust numerical methods for solving dynamical systems with multiple time scales

    Get PDF
    In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters

    Generalized averaged Gaussian quadrature and applications

    Get PDF
    A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

    MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

    Get PDF
    Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
    corecore