Most model reduction methods are space-only in that they reduce the spatial dimension of the solution but not the temporal one. These methods integrate an encoding of the state of the nonlinear dynamical system forward in time. We propose a space-time method -- one that solves a system of algebraic equations for the encoding of the trajectory, i.e., the solution on a time interval $[0,T]$. The benefit of this approach is that with the same total number of degrees of freedom, a space-time encoding can leverage spatiotemporal correlations to represent the trajectory far more accurately than a space-only one. We use spectral proper orthogonal decomposition (SPOD) modes, a spatial basis at each temporal frequency tailored to the structures that appear at that frequency, to represent the trajectory. These modes have a number of properties that make them an ideal choice for space-time model reduction. We derive an algebraic system involving the SPOD coefficients that represent the solution, as well as the initial condition and the forcing. The online phase of the method consists of solving this system for the SPOD coefficients given the initial condition and forcing. We test the model on a Ginzburg-Landau system, a $1 + 1$ dimensional nonlinear PDE. We find that the proposed method is $\sim 2$ orders of magnitude more accurate than POD-Galerkin at the same number of modes and CPU time for all of our tests. 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 space-only Petrov-Galerkin method.

翻译：大多数模型降阶方法仅针对空间维度，即降低解的维度而不处理时间维度。这些方法通过随时间向前积分来编码非线性动力系统的状态。我们提出了一种时空方法——该方法通过求解代数方程组来编码轨迹，即时间区间$[0,T]$上的解。这种方法的优势在于，在总自由度数量相同的情况下，时空编码能够利用时空相关性，比仅空间编码更准确地表示轨迹。我们采用谱本征正交分解（SPOD）模态——一种针对各时间频率下出现的结构定制的空间基函数——来表示轨迹。这些模态具有多种特性，使其成为时空模型降阶的理想选择。我们推导了包含表示解的SPOD系数、初始条件及外力的代数系统。该方法的在线阶段包括在给定初始条件和外力的情况下求解该系统的SPOD系数。我们在Ginzburg-Landau系统（一个$1 + 1$维非线性偏微分方程）上测试了该模型。结果表明，在所有测试中，当模态数量和CPU时间相同时，所提方法比POD-Galerkin方法的精度高出约2个数量级。实际上，该方法甚至显著优于将解投影到POD模态上的误差（该误差是任何仅空间Petrov-Galerkin方法误差的下界）。