Inference in Complex Stochastic Dynamic Environmental Models

现有的环境系统预报方法，如天气预报，往往基于确定性模型，通过一组必须在时间上向前积分的偏微分方程组来描述系统的时间演化。这忽略了这样一个事实，即为系统状态假设的计算模型和初始条件都只是对真实物理现实的近似。一种概率方法，通过将确定性模型替换为随机模型，以原则性的方式解决这些不确定性，应该允许进行优化预测。然而，涉及的变量数量巨大，使得对这种随机模型的准确处理成为一项难以计算的任务。最近通过集成具有不同初始条件的独立噪声系统的系综来近似环境系统概率分布的演化的尝试仅限于相当小的系综。本项目旨在开发新的计算方法，以便能够在大型复杂动态环境模型中进行近似概率推理。使用最初在统计物理学中发展并随后用于机器学习中的推理算法的想法，我们将开发对系统的概率分布(在空间和时间上)的近似，并在一类受控复杂性的近似内进行变量优化。包括部分测量(数据同化)对概率演变的影响的能力将使我们能够第一次估计未知的模型参数(如噪声项的精确特征)。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改善我们表示的稀疏性。使用完美的模型设置和蒙特卡罗方法(粒子过滤器)为小系统提供精确的解，我们将能够量化我们方法的精度，并将它们与其他常用的预报和数据同化方法，特别是4D变分方法和集合卡尔曼滤波进行比较。关键词：随机过程，动力系统，数据同化，概率模拟，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.