Exponential time differencing for matrix-valued dynamical systems

TL;DR

This paper extends exponential time differencing methods to matrix-valued dynamical systems, enabling stable and efficient integration of stiff equations common in control, machine learning, and PDE applications.

Contribution

The authors develop explicit matrix-ETD schemes for matrix evolution equations, including cases with non-commuting matrices, and prove their convergence and stability.

Findings

The schemes outperform competing methods in large-scale stiff systems.

Demonstrated efficiency on PDE-derived and matrix dynamics examples.

Achieved higher accuracy and stability in complex applications.

Abstract

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.