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. 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 (EKI). 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.
翻译:尽管许多系统的治理方程式(从最初的原则中得出)可能被视为已知的,但从数字上模拟它们描述的所有相互作用往往过于昂贵,因此研究人员常常寻求更简单的描述,描述复杂的现象,而不以数字方式解决所有相互作用的构成部分。在此背景下,Stochacistic 差异方程式(SDEs)自然地产生模型。通过实验和模拟,数据采集的增长为在许多学科中系统化地衍生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,SDEs和短期实际数据之间的不一致往往会造成问题。 SDEs与真实的动态之间的不兼容性可以通过从时间序列数据和基于这些要素的SDES学习参数中得出足够的统计数据加以解决。根据这种方法,我们将SDEs与从实际数据中充分获得的充足统计数据相匹配,作为反向问题,并表明这一反向问题可以通过使用堆积的Kalman模型(EKI)来加以解决。此外,我们创建了一个框架,通过引入分级、可重新定义的精确的精确度模型来进行漂浮化和扩散术语的精确度,然后在SDralimal的模型中,我们用一个模拟的模型来展示一个模拟的大规模的模型来验证。