A Lanczos-like method for non-autonomous linear ordinary differential equations

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

Strongly interacting electrons in solids are generically described by Hubbardtype models, and the impact of solar light can be modeled by an additional time-dependence. This yields a finite dimensional system of ordinary differential equations (ODE)s of Schrödinger type, which can be solved numerically by exponential time integrators of Magnus type. The efficiency may be enhanced by combining these with operator splittings. We will discuss several different approaches of employing exponential-based methods in conjunction with an adaptive Lanczos method for the evaluation of matrix exponentials…

二次降阶——高斯过程常微分方程框架用于大规模连续动力系统的推断

A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems

提出结合高斯过程与二次降阶的框架，实现复杂动力系统的高精度、稳定预测与不确定性量化。

A new Legendre polynomial-based approach for non-autonomous linear ODEs

We introduce a new method with spectral accuracy to solve linear non-autonomous ordinary differential equations (ODEs) of the kind $ \frac{d}{dt}\tilde{u}(t) = \tilde{f}(t) \tilde{u}(t)$, $\tilde{u}(-1)=1$, with $\tilde{f}(t)$ an analytic function. The method is based on a new expression for the solution $\tilde{u}(t)$ given in terms of a convolution-like operation, the $\star$-product. This expression is represented in a finite Legendre polynomial basis translating the initial problem into a matrix problem. An efficient procedure is proposed to approximate the Legendre coefficients of $\tilde…

通过半自主神经ODE实现动力系统通用逼近及其应用

Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications

本文提出半自主神经ODE，通过减少参数实现对动力系统的通用逼近，并验证其在捕捉ODE系统和传输方程动态方面的有效性。

保持常微分方程首次积分的离散梯度方法

Discrete gradient methods for preserving a first integral of an ordinary differential equation

本文研究离散梯度方法用于逼近自治常微分方程的解并保持其首次积分，证明了大类方法的存在性、局部唯一性和任意高阶构造，并针对二次首次积分提出了线性隐式高阶新方法。

Magnus integrators for linear and quasilinear delay differential equations

A procedure to numerically integrate non-autonomous linear delay differential equations is presented. It is based on the use of an spectral discretization of the delayed part to transform the original problem into a matrix linear ordinary differential equation which is subsequently solved with numerical integrators obtained from the Magnus expansion. The algorithm can be used in the periodic case to get both accurate approximations of the characteristic multipliers and the solution itself. In addition, it can be extended to deal with certain quasilinear delay equations.

非自治随机微分代数方程具有局部Lipschitz系数解的存在性与唯一性

Existence and uniqueness for the solutions of non-autonomous stochastic differential algebraic equations with locally Lipschitz coefficients

本文研究非自治随机微分代数方程在局部Lipschitz和单调系数下的适定性，通过转化为带代数约束的随机微分方程，证明了在$\mathcal{M}^p$空间中解的存在唯一性，并给出正则性结果。

The *-product approach for linear ODEs: a numerical study of the scalar case

Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the main ideas at the core of this approach in the simpler setting of a scalar ODE. Understanding the scalar case is fundamental since the method can be easily extended to the more challenging problem of systems of ODEs. Numerical examples illustrate the method's efficacy and its properties in the scalar case.

用于评估指数积分器中出现的phi函数的Krylov子空间算法

A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators

提出一种基于Krylov子空间的自适应算法，通过Arnoldi或Lanczos迭代计算矩阵指数及其相关phi函数作用在向量上的结果，用于高效求解线性常微分方程组，并在Matlab函数phipm中实现，数值实验表明其显著优于现有方法。

非线性有限维非自治动力系统的精确且最优二次化

Exact and optimal quadratization of nonlinear finite-dimensional non-autonomous dynamical systems

本文研究非自治、任意维非线性 ODE 的精确最优二次化，提出存在性理论与维度无关算法，并扩展 QBee，以更低维提升和降阶模型学习应用验证优势。

非局部随机神经元动力学的自主学习

Autonomous learning of nonlocal stochastic neuron dynamics

针对生物学合理的有色噪声驱动下神经元动力学PDF方程需闭合近似的问题，提出改进非局部大涡扩散率闭合和稀疏回归数据驱动闭合两种方法，在两类神经元模型上验证并计算了相关信息论量。