Strahan John, Finkel Justin, Dinner Aaron R, Weare Jonathan
Department of Chemistry and James Franck Institute, the University of Chicago, Chicago, IL 60637.
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139.
J Comput Phys. 2023 Sep 1;488. doi: 10.1016/j.jcp.2023.112152. Epub 2023 May 9.
Estimating the likelihood, timing, and nature of events is a major goal of modeling stochastic dynamical systems. When the event is rare in comparison with the timescales of simulation and/or measurement needed to resolve the elemental dynamics, accurate prediction from direct observations becomes challenging. In such cases a more effective approach is to cast statistics of interest as solutions to Feynman-Kac equations (partial differential equations). Here, we develop an approach to solve Feynman-Kac equations by training neural networks on short-trajectory data. Our approach is based on a Markov approximation but otherwise avoids assumptions about the underlying model and dynamics. This makes it applicable to treating complex computational models and observational data. We illustrate the advantages of our method using a low-dimensional model that facilitates visualization, and this analysis motivates an adaptive sampling strategy that allows on-the-fly identification of and addition of data to regions important for predicting the statistics of interest. Finally, we demonstrate that we can compute accurate statistics for a 75-dimensional model of sudden stratospheric warming. This system provides a stringent test bed for our method.
估计事件的可能性、发生时间和性质是对随机动力系统进行建模的主要目标。当与解析基本动力学所需的模拟和/或测量时间尺度相比，该事件较为罕见时，基于直接观测进行准确预测就变得具有挑战性。在这种情况下，一种更有效的方法是将感兴趣的统计量表示为费曼 - 卡茨方程（偏微分方程）的解。在此，我们开发了一种通过在短轨迹数据上训练神经网络来求解费曼 - 卡茨方程的方法。我们的方法基于马尔可夫近似，但避免了对基础模型和动力学的假设。这使得它适用于处理复杂的计算模型和观测数据。我们使用一个便于可视化的低维模型来说明我们方法的优势，并且该分析促使我们采用一种自适应采样策略，该策略允许即时识别对预测感兴趣的统计量重要的区域并向这些区域添加数据。最后，我们证明我们能够为平流层突然变暖的75维模型计算准确的统计量。该系统为我们的方法提供了一个严格的测试平台。
