33 research outputs found
Hierarchical deep learning-based adaptive time-stepping scheme for multiscale simulations
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
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
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
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
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
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
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