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：数据驱动降阶模型常难以准确预测高维非线性动力系统，这类系统在低方差坐标方向上具有敏感性，因为此类坐标常被截断（例如通过本征正交分解、核主成分分析及自编码器）。在剪切主导的流体流动中频繁出现此类系统，非正态性对扰动增长起重要作用。为解决这些问题，我们借鉴活动子空间思想，通过平衡系统敏感性的伴随信息与轨迹状态方差，寻找低维坐标系统进行模型降阶。所提出的方法——称为基于伴随快照的协方差平衡降阶（CoBRAS）——类似于用状态与伴随梯度协方差矩阵替代系统Gramian矩阵的平衡截断方法，且遵循相同关键变换律。此处提取的坐标与斜投影相关联，可用于构建Petrov-Galerkin降阶模型。我们提供类似于平衡本征正交分解的高效快照计算方案，进而发现降阶坐标可仅依赖状态与梯度样本的内积计算，通过将内积替换为核函数来获取丰富非线性坐标。在此类坐标下，可使用回归方法学习降阶模型。我们在一个简单但具有挑战性的三维系统及具有$10^5$状态变量的非线性轴对称射流流动模拟中演示这些技术，并与多种其他方法进行对比。