Model reduction techniques have emerged as a powerful paradigm across different fronts of scientific computing. Despite their success, the provided tools and methodologies remain limited if high-dimensional dynamical systems subject to initial uncertainty and/or stochastic noise are encountered; in particular if rare events are of interest. We address this open challenge by borrowing ideas from Mori-Zwanzig formalism and Chorin's optimal prediction method. The novelty of our work lies on employing time-dependent optimal projection of the dynamic on a desired set of resolved variables. We show several theoretical and numerical properties of our model reduction approach. In particular, we show that the devised surrogate trajectories are consistent with the probability flow of the full-order system. Furthermore, we identify the measure underlying the projection through polynomial chaos expansion technique. This allows us to efficiently compute the projection even for trajectories that are initiated on low probability events. Moreover, we investigate the introduced model-reduction error of the surrogate trajectories on a standard setup, characterizing the convergence behaviour of the scheme. Several numerical results highlight the computational advantages of the proposed scheme in comparison to Monte-Carlo and optimal prediction method. Through this framework, we demonstrate that by tracking the measure along with the consistent projection of the dynamic we are able to access accurate estimates of different statistics including observables conditional on a given initial configuration.

翻译：模型降阶技术已成为科学计算不同领域中的一种强大范式。尽管取得了成功，当遇到受初始不确定性和/或随机噪声影响的高维动力系统时，特别是当关注罕见事件时，现有工具和方法仍存在局限。我们通过借鉴Mori-Zwanzig形式体系和Chorin最优预测方法的思想来应对这一开放挑战。本工作的创新之处在于采用随时间变化的最优投影，将动力学投影到一组期望的解析变量上。我们展示了该模型降阶方法的若干理论与数值性质。特别地，我们证明了所设计的代理轨迹与全阶系统的概率流保持一致。此外，我们通过多项式混沌展开技术识别了投影所基于的测度。这使得我们能够高效计算投影，即使对于始于低概率事件的轨迹也是如此。进一步，我们在标准设置下研究了代理轨迹引入的模型降阶误差，刻画了该方案的收敛行为。多项数值结果突显了所提方案相较于蒙特卡洛方法和最优预测方法的计算优势。通过该框架，我们证明通过追踪测度并结合动力学的相容投影，能够准确估计包括依赖于给定初始构型的可观测量在内的多种统计量。