Data-driven reduced-order models often fail to make accurate forecasts of high-dimensional nonlinear dynamical systems that are sensitive along coordinates with low-variance because such coordinates are often truncated, e.g., by proper orthogonal decomposition, kernel principal component analysis, and autoencoders. Such systems are encountered frequently in shear-dominated fluid flows where non-normality plays a significant role in the growth of disturbances. In order to address these issues, we employ ideas from active subspaces to find low-dimensional systems of coordinates for model reduction that balance adjoint-based information about the system's sensitivity with the variance of states along trajectories. The resulting method, which we refer to as covariance balancing reduction using adjoint snapshots (CoBRAS), is analogous to balanced truncation with state and adjoint-based gradient covariance matrices replacing the system Gramians and obeying the same key transformation laws. Here, the extracted coordinates are associated with an oblique projection that can be used to construct Petrov-Galerkin reduced-order models. We provide an efficient snapshot-based computational method analogous to balanced proper orthogonal decomposition. This also leads to the observation that the reduced coordinates can be computed relying on inner products of state and gradient samples alone, allowing us to find rich nonlinear coordinates by replacing the inner product with a kernel function. In these coordinates, reduced-order models can be learned using regression. We demonstrate these techniques and compare to a variety of other methods on a simple, yet challenging three-dimensional system and a nonlinear axisymmetric jet flow simulation with $10^5$ state variables.
翻译:数据驱动的降阶模型经常在低方差坐标的敏感度截断时无法准确预测高维度的非线性动力系统,例如可以通过正交分解、核主成分分析和自动编码器进行截断。在剪切主导的流体流中经常遇到此类系统,其中非正常性在扰动增长中起着重要作用。为了解决这些问题,我们采用主动子空间的思想,找到基于状态灵敏度和沿轨迹状态变量方差平衡的低维坐标系来进行模型减缩。所得到的方法,被我们称之为利用伴随快照平衡协方差减缩(CoBRAS)方法,类似于平衡截断,将状态和伴随梯度协方差矩阵替代系统Gramian,并遵守相同的关键转化法则。此处,提取的坐标和随后产生的满足基于Sylvester方程的非对称矩阵问题,这些矩阵可以用于构造Petrov-Galerkin降阶模型。我们提供了一种类似于平衡正交分解的有效基于快照的计算方法,同时,这也导致观察到只需要基于状态和梯度样本的内积即可计算所降维的坐标,从而允许我们通过将内积替换为核函数来寻找富有的非线性坐标。在这些坐标中,可以使用回归的方法来学习降阶模型。我们在一个简单但具有挑战性的三维系统和一个具有$10^5$个状态变量的非线性轴对称射线流模拟中展示了这些技术,并与其他方法进行了比较。