Matrix evolution equations occur in many applications, such as dynamical
Lyapunov/Sylvester systems or Riccati equations in optimization and stochastic
control, machine learning or data assimilation. In many cases, their tightest
stability condition is coming from a linear term. Exponential time differencing
(ETD) is known to produce highly stable numerical schemes by treating the
linear term in an exact fashion. In particular, for stiff problems, ETD methods
are a method of choice. We propose an extension of the class of ETD algorithms
to matrix-valued dynamical equations. This allows us to produce highly
efficient and stable integration schemes. We show their efficiency and
applicability for a variety of real-world problems, from geophysical
applications to dynamical problems in machine learning.

矩阵演化方程是许多应用中的核心，扩展了 ETD 算法类的 ETD 算法是高效且稳定的数值整合方案，并适用于多种现实世界问题。

矩阵值动态系统的指数时间离散化

Exponential time differencing for matrix-valued dynamical systems

Modeling of conservative systems with neural networks is an area of active
research. A popular approach is to use Hamiltonian neural networks (HNNs) which
rely on the assumptions that a conservative system is described with Hamilton's
equations of motion. Many recent works focus on improving the integration
schemes used when training HNNs. In this work, we propose to enhance HNNs with
an estimation of a continuous-time trajectory of the modeled system using an
additional neural network, called a deep hidden physics model in the
literature. We demonstrate that the proposed integration scheme works well for
HNNs, especially with low sampling rates, noisy and irregular observations.

本文提出了一种使用提高了的积分方案的 Hamilton 神经网络，结合使用深度隐藏的物理模型来对保守系统进行数值模拟的方法，可以成功处理低采样率、嘈杂和不准确观测值。

用神经网络学习哈密顿系统的轨迹

Learning Trajectories of Hamiltonian Systems with Neural Networks

When simulating trajectories by integrating time-continuous car-following
models, standard integration schemes such as the forth-order Runge-Kutta method
(RK4) are rarely used while the simple Euler's method is popular among
researchers. We compare four explicit methods: Euler's method, ballistic
update, Heun's method (trapezoidal rule), and the standard forth-order RK4. As
performance metrics, we plot the global discretization error as a function of
the numerical complexity. We tested the methods on several time-continuous
car-following models in several multi-vehicle simulation scenarios with and
without discontinuities such as stops or a discontinuous behavior of an
external leader. We find that the theoretical advantage of RK4 (consistency
order~4) only plays a role if both the acceleration function of the model and
the external data of the simulation scenario are sufficiently often
differentiable. Otherwise, we obtain lower (and often fractional) consistency
orders. Although, to our knowledge, Heun's method has never been used for
integrating car-following models, it turns out to be the best scheme for many
practical situations. The ballistic update always prevails Euler's method
although both are of first order.

本文以多车辆模拟场景为背景，比较 Euler 方法、Ballistic Update、梯形规则方法和标准四阶 RK4 方法的全局分段误差，并发现在许多实际情况下，虽然标准四阶 RK4 方法也存在一定优势，但 Heun 方法可以成为合适的选择。最后，球式更新始终胜过 Euler 方法。