常微分方程外推法的性能评估：基于无误差变换的方法

Parallelizing Explicit and Implicit Extrapolation Methods for Ordinary Differential Equations

Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using extrapolation methods which exploit "parallelism within the method" so that arbitrary user ODEs can be parallelized. We describe the specific choices made in the implementation of the explicit and implicit extrapolation methods which allow for generating low overhead static schedules to then exploit with optimized multi-threaded implementations. We demonstrate that wh…

基于自编码器的降阶神经常微分方程的误差传播分析

Analysis of Error Propagation in Autoencoder-Based Reduced-Order Neural Ordinary Differential Equations

本文针对基于自编码器的降阶神经常微分方程，提出路径积分恒等式分析其误差传播，通过多步雅可比范数区分传播机制，实验揭示了Burgers与Gray-Scott系统的不同误差演化模式。

常微分方程的多速率方法

Multirate methods for ordinary differential equations

本文综述了多速率方法，通过插值/外推耦合变量或推广经典单速率方法，以适应不同时间尺度并降低计算成本。

Inter/extrapolation-based multirate schemes -- a dynamic-iteration perspective

Multirate behavior of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) is characterized by widely separated time constants in different components of the solution or different additive terms of the right-hand side. Here, classical multirate schemes are dedicated solvers, which apply (e.g.) micro and macro steps to resolve fast and slow changes in a transient simulation accordingly. The use of extrapolation and interpolation procedures is a genuine way for coupling the different parts, which are defined on different time grids. This paper contains for the first…

A Non-Iterative Transformation Method for an Extended Blasius Problem

In this paper we define a non-iterative transformation method for an Extended Blasius Problem. The original non-iterative transformation method, which is based on scaling invariance properties, was defined for the classical Blasius problem by Töpfer in 1912. This method allows us to solve numerically a boundary value problem by solving a related initial value problem and then rescaling the obtained numerical solution. In recent years, we have seen applications of the non-iterative transformation method to several problems of interest. The obtained numerical results, are improved by both a mesh…

Taylor Polynomials in High Arithmetic Precision as Universal Approximators

Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is demonstrated, by utilizing partial sums of Taylor series in high arithmetic precision. In particular, the proposed method is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method is based on the utilization of Taylor polynomials, by exploit…