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 such problems, the dominant stability restriction is imposed by a stiff linear term, making standard explicit integrators impractical. 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 the methods of choice. We extend ETD to matrix-valued evolution equations of the form $\dot Q = LQ + QR + N(Q,t)$ by deriving explicit matrix-ETD (METD) schemes. When $L$ and $R$ commute, we construct an explicit $p$-th order METD$p$ family and prove order-$p$ global convergence under standard assumptions; for the non-commuting case, we develop a Baker-Campbell-Hausdorff (BCH)-based extension. This allows us to produce highly efficient and stable integration schemes. We demonstrate efficiency and applicability on stiff PDE-derived and large-scale matrix dynamics, including an Allen-Cahn system, turbulent jet fluctuation statistics, and continuous graph neural networks. We further show that the scheme is more accurate, stable, and efficient than competing schemes in large-scale high-rank stiff systems.

翻译：矩阵演化方程在许多应用中出现，例如动力李雅普诺夫/西尔维斯特系统或优化与随机控制中的里卡蒂方程、机器学习或数据同化。在许多此类问题中，主要的稳定性限制由刚性线性项施加，使得标准显式积分器不实用。指数时间差分（ETD）通过精确处理线性项，可产生高度稳定的数值格式。特别是对于刚性问题，ETD方法是首选方法。我们将ETD推广到形式为$\dot Q = LQ + QR + N(Q,t)$的矩阵值演化方程，通过推导显式矩阵-ETD（METD）格式。当$L$和$R$可交换时，我们构造了显式$p$阶METD$p$族，并在标准假设下证明了$p$阶全局收敛性；对于不可交换情形，我们开发了基于Baker-Campbell-Hausdorff（BCH）的扩展。这使我们能够产生高效且稳定的积分格式。我们在刚性偏微分方程导出的大规模矩阵动力学上展示了效率和适用性，包括Allen-Cahn系统、湍流射流涨落统计和连续图神经网络。我们进一步证明，该格式在大规模高秩刚性系统中比竞争格式更精确、稳定和高效。