Learning stochastic closures using ensemble Kalman inversion
Learning stochastic closures using ensemble Kalman inversion
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DOI：
10.1093/imatrm/tnab003
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发表时间：
2020-04
影响因子：
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通讯作者：
T. Schneider;A. Stuart;Jin-Long Wu
中科院分区：
文献类型：
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作者：
T. Schneider;A. Stuart;Jin-Long Wu
尽管许多系统的控制方程，当从第一原理得到时，可能被认为是已知的，但要对它们所描述的所有相互作用进行数值模拟往往代价太高。因此，研究人员经常寻求更简单的描述，描述复杂的现象，而不是用数字来解析所有相互作用的组成部分。在这种背景下，随机微分方程(SDE)作为模型应运而生。数据采集的增长，无论是通过实验还是通过模拟，都为在许多学科中系统地推导SDE模型提供了机会。然而，在应用标准统计方法进行参数估计时，短时间尺度下SDE与实际数据之间的不一致往往会造成问题。可以通过从时间序列数据中获得足够的统计量和基于这些统计量的SDE的学习参数来解决SDE与实际数据之间的不兼容问题。在这里，我们研究从时间平均计算的足够的统计数据，我们展示的方法可以导致对各种问题的足够的统计数据，并且具有消除匹配轨迹的需要的次要好处。按照这种方法，我们将SDE对真实数据的充分统计数据的拟合表示为一个反问题，并证明了这个反问题可以通过集合卡尔曼反演来解决。此外，通过引入未知函数的分层、可细化的参数化，使用高斯过程回归，我们创建了用于漂移和扩散项的非参数学习的框架。我们展示了所提出的SDE模型的拟合方法，首先是在一个带有噪声的Lorenz‘63模型的模拟研究中，然后是在其他应用中，包括大气科学中出现的确定性混沌系统的降维，气候动力学中的大规模模式建模，以及分子动力学中出现的关键观测的简化模型。结果证实，所提出的方法提供了一种稳健和系统的方法来将SDE模型与实际数据进行拟合。
Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore, researchers often seek simpler descriptions that describe complex phenomena without numerically resolving all the interacting components. Stochastic differential equations (SDEs) arise naturally as models in this context. The growth in data acquisition, both through experiment and through simulations, provides an opportunity for the systematic derivation of SDE models in many disciplines. However, inconsistencies between SDEs and real data at short time scales often cause problems, when standard statistical methodology is applied to parameter estimation. The incompatibility between SDEs and real data can be addressed by deriving sufficient statistics from the time-series data and learning parameters of SDEs based on these. Here, we study sufficient statistics computed from time averages, an approach that we demonstrate to lead to sufficient statistics on a variety of problems and that has the secondary benefit of obviating the need to match trajectories. Following this approach, we formulate the fitting of SDEs to sufficient statistics from real data as an inverse problem and demonstrate that this inverse problem can be solved by using ensemble Kalman inversion. Furthermore, we create a framework for non-parametric learning of drift and diffusion terms by introducing hierarchical, refinable parameterizations of unknown functions, using Gaussian process regression. We demonstrate the proposed methodology for the fitting of SDE models, first in a simulation study with a noisy Lorenz ’63 model, and then in other applications, including dimension reduction in deterministic chaotic systems arising in the atmospheric sciences, large-scale pattern modeling in climate dynamics and simplified models for key observables arising in molecular dynamics. The results confirm that the proposed methodology provides a robust and systematic approach to fitting SDE models to real data.