We propose a space-time reduced-order model (ROM) for nonlinear dynamical systems, building upon previous work on linear systems. Whereas most ROMs are space-only in that they reduce only the spatial dimension of the state, the proposed method leverages an efficient encoding of the entire trajectory of the state on the time interval $[0,T]$, enabling significant additional reduction. Trajectories are encoded using SPOD modes, a spatial basis at each temporal frequency tailored to the structures that appear at that frequency. These modes have a number of properties that make them an ideal choice for space-time model reduction, including separability and near-optimality for long trajectories. We derive a system of algebraic equations involving the SPOD coefficients, forcing, and initial condition by projecting an implicit solution of the governing equations onto the set of SPOD modes in a space-time inner product. We therefore refer to the method as spectral solution operator projection (SSOP). The online phase of SSOP comprises solving this system for the SPOD coefficients, given the initial condition and forcing. We find that SSOP gives two orders of magnitude lower error than POD-Galerkin projection at the same number of modes and CPU time across a suite of tests, including ones that use out-of-sample forcings and affine parameter variation. In fact, the method is substantially more accurate even than the projection of the solution onto the POD modes, which is a lower bound for the error of any method based on a linear space-only encoding of the state.

翻译：本文提出了一种针对非线性动力系统的时空降阶模型，该模型建立在先前线性系统研究的基础上。与大多数仅降低状态空间维度的空间降阶模型不同，本方法通过高效编码状态在时间区间$[0,T]$上的完整轨迹，实现了显著的额外降阶。轨迹编码采用SPOD模态，即针对各时间频率下出现的结构定制的空间基函数。这些模态具备可分性和对长轨迹的近似最优性等特性，使其成为时空模型降阶的理想选择。通过将控制方程的隐式解投影到SPOD模态构成的时空内积空间，我们推导出包含SPOD系数、外力和初始条件的代数方程组。因此，该方法被称为谱解算子投影法。SSOP的在线阶段包括在给定初始条件和外力的情况下求解该方程组以获得SPOD系数。通过一系列测试（包括使用样本外激励和仿射参数变化的场景）发现，在相同模态数量和CPU时间下，SSOP的误差比POD-Galerkin投影法低两个数量级。实际上，该方法甚至显著优于将解投影到POD模态的误差下限——该下限是所有基于线性空间状态编码方法的误差基准。