In this work, the modally equivalent perturbed system (MEPS) which was originally developed for finding the parametrically rich solution space of linear time‐invariant systems is modified for time‐varying cases and applied to find the characteristically rich nonlinear solution space given arbitrary initial or boundary conditions, or system inputs. An integral form of the non‐Hamiltonian Liouville equation is derived such that a rich ensemble average of its solutions covers a broad range of the modal space when a maximum uncertainty is present in the solutions. The MEPS degenerates the integrated Liouville equation into a linear differential equation with the Gauge Modal Invariance, a newly found field property that allows extending the application beyond the initial conditions or impulse inputs, making it possible to calculate the rich set of basis modes by taking snapshots of the linear responses at a considerably low computational cost. The proposed theory and algorithm are demonstrated using a computational model of a two‐dimensional incompressible, viscous flow at low Reynolds numbers. It is shown that the basis modes obtained herein, when used in conjunction with a low dimensional modeling, reproduce time simulation results very accurately for a wide range of Reynolds numbers and boundary conditions.

中文翻译：

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寻找特征丰富的非线性解空间：一种统计力学方法

在这项工作中，最初为发现线性时不变系统的参数丰富解空间而开发的模态等效摄动系统（MEPS）被修改为用于时变情况，并应用于给定任意初始值或初始值的特征丰富的非线性解空间。边界条件或系统输入。推导出非哈密顿型Liouville方程的积分形式，这样，当解决方案中存在最大不确定性时，其解决方案的丰富整体平均值便会覆盖广泛的模态空间。MEPS使用Gauge Modal Invariance（量规模态不变性）将集成的Liouville方程退化为线性微分方程，Gauge Modal Invariance是一种新发现的磁场特性，可以将应用范围扩展到初始条件或脉冲输入之外，通过以相当低的计算成本拍摄线性响应的快照，可以计算出丰富的基本模式集。利用二维不可压缩粘性流在低雷诺数下的计算模型论证了所提出的理论和算法。可以看出，当与低维建模结合使用时，本文获得的基本模式可以在宽范围的雷诺数和边界条件下非常精确地再现时间模拟结果。