Inference in Complex Stochastic Dynamic Environmental Models

目前对环境系统（如天气）的预测方法通常基于确定性模型，这种模型通过一组偏微分方程来描述系统的时间演化，这些偏微分方程必须在时间上向前积分。这忽略了这样一个事实，即计算模型和系统状态的初始条件都只是真实物理现实的近似值。一种概率方法，通过用随机模型取代确定性模型，以原则性的方式解决这些不确定性，应该允许优化预测。然而，涉及的大量变量使得精确处理这种随机模型成为一项计算上难以处理的任务。最近试图通过对具有不同初始条件的独立噪声系统的集成来近似环境系统概率分布的演化，但仅限于相当小的集成。本项目旨在开发新的计算方法，以便在大型复杂动态环境模型中实现近似概率推理。使用最初在统计物理学中开发的思想，随后用于机器学习中的推理算法，我们将开发系统概率分布（在空间和时间上）的近似值，这些近似值在一类控制复杂性的近似值中进行变分优化。包含部分测量（数据同化）对概率演化的影响的能力也将使我们第一次能够估计未知的模型参数（如噪声项的精确特征）。我们将扩展将4D VAR数据同化方法理解为哈密顿形式的控制问题的最新进展，以协助定义噪声过程并利用对称性来改善我们的表示的稀疏性。利用完美的模型设置和蒙特卡罗方法（粒子滤波）为小系统提供精确的解决方案，我们将能够量化我们方法的准确性，并将它们与其他常用的预测和数据同化方法，特别是4D变分方法和集合卡尔曼滤波进行对比。关键词：随机过程，动力系统，数据同化，概率建模，高斯过程，变分方法，贝叶斯，模型误差，控制问题。

Present prediction methods for environmental systems, such as the weather, are often based on deterministic models which describe the time evolution of the system by a set of partial differential equations which have to be integrated forward in time. This ignores the fact that both the computational models and the initial conditions assumed for the system's state are only approximations to the true physical reality. A probabilistic approach, which would address these uncertainties in a principled way by replacing the deterministic model by a stochastic one, should allow for optimised predictions. However, the huge number of variables involved renders the exact treatment of such stochastic models a computationally intractable task. Recent attempts to approximate the evolution of probability distributions for environmental systems by integrating an ensemble of independent noisy systems with different initial conditions are restricted to rather small ensembles. The present project aims at developing new computational methods to enable approximate probabilistic inference in large complex dynamical environmental models. Using ideas originally developed in statistical physics and subsequently used for inference algorithms in machine learning, we will develop approximations to the system's probability distribution (in space and time) which are variationally optimised within a class of approximations of controlled complexity. The ability to include the effect of partial measurements (data assimilation) on the evolution of probabilities will allow us also to estimate unknown model parameters (like the precise characteristics of the noise terms) for the first time. We will extend recent developments in understanding 4D VAR data assimilation methods as control problems in Hamiltonian form, to assist in the definition of the noise processes and in exploiting symmetries to improve the sparsity of our representation. Using perfect model settings, and Monte Carlo methods (particle filters) to provide exact solutions for small systems, we will be able to quantify the accuracy of our methods and contrast them with other commonly used forecasting and data assimilation methods, especially 4D variational methods and the ensemble Kalman filter.Keywords: stochastic processes, dynamical systems, data assimilation, probabilistic modelling, Gaussian processes, variational methods, Bayesian, model error, control problems.