We propose Comprehensive Robust Dynamic Mode Decomposition (CR-DMD), a novel framework that robustifies the entire DMD process - from mode extraction to dimensional reduction - against mixed noise. Although standard DMD widely used for uncovering spatio-temporal patterns and constructing low-dimensional models of dynamical systems, it suffers from significant performance degradation under noise due to its reliance on least-squares estimation for computing the linear time evolution operator. Existing robust variants typically modify the least-squares formulation, but they remain unstable and fail to ensure faithful low-dimensional representations. First, we introduce a convex optimization-based preprocessing method designed to effectively remove mixed noise, achieving accurate and stable mode extraction. Second, we propose a new convex formulation for dimensional reduction that explicitly links the robustly extracted modes to the original noisy observations, constructing a faithful representation of the original data via a sparse weighted sum of the modes. Both stages are efficiently solved by a preconditioned primal-dual splitting method. Experiments on fluid dynamics datasets demonstrate that CR-DMD consistently outperforms state-of-the-art robust DMD methods in terms of mode accuracy and fidelity of low-dimensional representations under noisy conditions.

翻译：我们提出综合鲁棒动态模态分解（CR-DMD）这一新颖框架，该框架能够增强整个DMD流程——从模态提取到降维——对混合噪声的鲁棒性。尽管标准DMD被广泛用于揭示时空模式并构建动力系统的低维模型，但由于其依赖最小二乘估计来计算线性时间演化算子，在噪声环境下性能会显著下降。现有的鲁棒变体通常修改最小二乘公式，但它们仍然不稳定，且无法确保忠实的低维表示。首先，我们引入一种基于凸优化的预处理方法，旨在有效去除混合噪声，实现准确稳定的模态提取。其次，我们提出一种新的降维凸优化公式，将鲁棒提取的模态与原始含噪观测显式关联，通过模态的稀疏加权和构建原始数据的忠实表示。两个阶段均通过预条件原对偶分裂方法高效求解。在流体动力学数据集上的实验表明，在噪声条件下，CR-DMD在模态精度和低维表示保真度方面始终优于最先进的鲁棒DMD方法。