Semiparametric Methods for Data Assimilation and Uncertainty Quantification

数据同化和不确定性量化的半参数方法

基本信息

批准号：
2006808
负责人：
Tyrus Berry
金额：
$ 23.37万
依托单位：
George Mason University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006808&HistoricalAwards=false
关键词：
Semiparametric Methods Data Assimilation Uncertainty

项目摘要

There is a growing demand in many scientific disciplines for efficient tools to automatically learn models and make predictions from limited noisy observations. For these predictions to be actionable, they must also have quantifiable uncertainty, and be robust to model misspecification. This is particularly relevant in light of events such as the COVID-19 pandemic, where models have to be constantly adapted to include new phenomena such as unreported and asymptomatic cases and constantly evolving social distancing rules and compliance. Other applications include large complex systems such as weather forecasting and social network dynamics where first-principles models are powerful but have difficulty capturing the full range of phenomena involved. The semiparametric framework will help address the growing problem of un-modeled phenomena by allowing existing models to be automatically merged with model-free methods that leverage data to learn a correction to the model in order to match the observed data. The new tools will allow application to a class of high dimensional problems with spatial structure, such as geosystems problems, social networks, and global disease dynamics. Beyond improving forecasting, the semiparametric approach will include accurate uncertainty quantification, which is critical in these application domains. The investigator will train a graduate student and undergraduate students who will be able to carry this research forward, as well as developing and disseminating this key expertise. These students will learn to apply both state-of-the-art and the newly developed methods which will prepare them for future work in applied and computational mathematics.The investigator will develop semiparametric modeling techniques that optimally leverage the strengths of parametric (model based) and nonparametric (model-free or data-driven) methods. Specifically, the semiparametric framework allows the flexible nonparametric models to fill in the gaps and correct the low-dimensional model error in a parametric model. The framework employs an ensemble of states in the parametric model to represent the uncertainty in a forecast or state estimate, while a full probability distribution is estimated for the nonparametric model. At each filtering or forecasting step, the ensemble is updated by sampling individual corrections from the model error distribution estimated by the nonparametric model. These sampled corrections will automatically correct biases in the model and inflate the uncertainty when necessary in order to match reality. The evolution of the nonparametric model will typically need to be conditional to the high-dimensional state of the parametric model, which current methods to do not allow. In other words, information must flow in both directions: the nonparametric model corrects the parametric model, but is also informed by the current state of the parametric model. In order to overcome this crucial challenge, supervised dimensionality reduction techniques will be combined with a novel method of learning mappings between non-diffeomorphic spaces. This will allow a Bayesian update of the nonparametric state estimate based on the learned projection of the parametric state. The research includes a novel higher order unscented ensemble forecast that will form the basis for a higher order Kalman filter. These advances will make the best use of available computation resources, since the higher order ensemble forecasting and filtering methods can scale up from small to large ensembles as resources allow. The higher order methods will improve accuracy and uncertainty quantification by estimating higher order moments of the state estimate and the forecast. For the ensemble forecast, a novel multivariate quadrature method will be applied that uses rank-1 tensor decompositions of the higher moments as quadrature nodes. For the Kalman update, higher order equations will be used based on a maximum entropy closure of the moment equations derived from the Kushner equation (which fully describes the true solution). The advances will effectively use data to learn a model-free correction to a parametric model, simultaneously alleviating model error and the curse-of-dimensionality.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

许多科学学科对有效工具的需求不断增长，以自动学习模型并从有限的嘈杂观察中做出预测。为了使这些预测是可行的，它们还必须具有可量化的不确定性，并且可以建模错误。这是根据Covid-19大流行等事件尤其重要的，必须不断适应模型，以包括新现象，例如未报告和无症状案件以及不断发展的社会疏远规则和合规性。其他应用程序包括大型复杂系统，例如天气预报和社交网络动态，其中第一原则模型具有强大的功能，但很难捕获所涉及的全部现象。半参数框架将通过允许将现有模型与无模型的方法合并，以利用数据来学习对模型的校正以匹配观察到的数据，从而有助于解决未建模现象的日益增长的问题。这些新工具将允许在空间结构（例如地理系统问题，社交网络和全球疾病动态）等一类高维问题中应用。除了改善预测外，半参数方法还将包括准确的不确定性量化，这在这些应用领域至关重要。调查人员将培训一名研究生和本科生，他们将能够提出这项研究，并开发和传播这一关键专业知识。这些学生将学会应用最新的方法和新开发的方法，这些方法将使它们为应用和计算数学的未来工作做好准备。和非参数（无模型或数据驱动）方法。具体而言，半参数框架允许灵活的非参数模型填补空白并纠正参数模型中的低维模型误差。该框架采用参数模型中的状态集合来表示预测或状态估计中的不确定性，而对非参数模型则估算了完全的概率分布。在每个过滤或预测步骤中，通过从非参数模型估计的模型误差分布中对单个校正进行采样来更新集合。这些采样校正将自动纠正模型中的偏差，并在必要时膨胀不确定性以匹配现实。非参数模型的演变通常需要以参数模型的高维状态为条件，而当前方法不允许使用。换句话说，信息必须以两个方向流动：非参数模型纠正了参数模型，但也以参数模型的当前状态告知。为了克服这一至关重要的挑战，监督的降维技术将与一种新型的学习映射方法相结合，可在非呈型肌形空间之间学习映射。这将允许基于参数状态的学习预测对非参数状态估计的贝叶斯更新。这项研究包括一个新型的高阶非式集合预测，这将构成高级卡尔曼滤波器的基础。这些进步将充分利用可用的计算资源，因为高阶合奏预测和过滤方法可以随着资源允许的范围从小到大的合奏扩展。高阶方法将通过估计状态估计和预测的高阶力矩来提高准确性和不确定性量化。对于合奏的预测，将应用一种新型的多元正交方法，该方法将较高矩的rank-1张量分解作为正交节点。对于Kalman更新，将基于从Kushner方程得出的矩方程的最大熵闭合（完全描述了真实的解决方案）。这些进步将有效地使用数据来学习对参数模型的无模型校正，同时减轻模型错误和差异性的诅咒。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子值得评估的支持的。和更广泛的影响审查标准。