We propose a procedure for the numerical approximation of invariance equations arising in the moment matching technique associated with reduced-order modeling of high-dimensional dynamical systems. The Galerkin residual method is employed to find an approximate solution to the invariance equation using a Newton iteration on the coefficients of a monomial basis expansion of the solution. These solutions to the invariance equations can then be used to construct reduced-order models. We assess the ability of the method to solve the invariance PDE system as well as to achieve moment matching and recover a system's steady-state behaviour for linear and nonlinear signal generators with system dynamics up to $n=1000$ dimensions.

翻译：本文提出了一种数值逼近方法，用于求解高维动力系统降阶建模中矩匹配技术所对应的不变性方程。该方法采用伽辽金残差法，通过对解在单项式基展开系数进行牛顿迭代，求得不变性方程的近似解。所得不变性方程的解可用于构建降阶模型。我们评估了该方法在求解不变性偏微分方程组、实现矩匹配以及恢复系统稳态行为方面的能力，测试对象包括线性和非线性信号发生器，其系统维度高达$n=1000$。