Inference in Complex Stochastic Dynamic Environmental Models

目前的环境系统预测方法，如天气，往往是基于确定性模型，它描述了一组偏微分方程的时间演化的系统，必须在时间上向前整合。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对真实物理现实的近似。概率性方法将通过用随机性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，所涉及的变量的巨大数量使得这种随机模型的精确处理在计算上是一个棘手的任务。最近的尝试，以近似的概率分布的环境系统的演化，通过整合一个合奏的独立的嘈杂的系统与不同的初始条件被限制到相当小的合奏。本项目旨在开发新的计算方法，使大型复杂的动态环境模型的近似概率推理。使用最初在统计物理学中开发并随后用于机器学习中的推理算法的想法，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。包括部分测量（数据同化）对概率演变的影响的能力，也将使我们能够首次估计未知的模型参数（如噪声项的精确特性）。我们将扩展在理解4D VAR数据同化方法作为哈密顿形式的控制问题方面的最新进展，以帮助定义噪声过程并利用对称性来提高我们表示的稀疏性。使用完美的模型设置和蒙特卡罗方法（粒子滤波器）为小系统提供精确解，我们将能够量化我们的方法的准确性，并将其与其他常用的预报和数据同化方法进行对比，特别是四维变分方法和集合卡尔曼滤波。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.