Semiparametric Methods for Data Assimilation and Uncertainty Quantification

在许多科学学科中，对自动学习模型并从有限的噪声观测中进行预测的有效工具的需求不断增长。为了使这些预测具有可操作性，它们还必须具有可量化的不确定性，并且对模型错误设定具有鲁棒性。鉴于COVID-19大流行等事件，这一点尤其重要，因为模型必须不断调整，以包括新现象，如未报告和无症状病例，以及不断变化的社交距离规则和合规性。其他应用包括大型复杂系统，如天气预报和社交网络动态，其中第一原理模型功能强大，但难以捕捉所涉及的全部现象。半参数框架将有助于解决日益严重的未建模现象问题，允许现有模型自动与无模型方法合并，利用数据学习模型校正，以匹配观察到的数据。新的工具将允许应用到一类具有空间结构的高维问题，如地理系统问题，社交网络和全球疾病动力学。除了改进预测外，半参数方法还将包括准确的不确定性量化，这在这些应用领域中至关重要。研究人员将培训一名研究生和一名本科生，他们将能够推进这项研究，并开发和传播这一关键专业知识。这些学生将学习应用最先进的和新开发的方法，这将为他们在应用和计算数学的未来工作做好准备。研究人员将开发半参数建模技术，最佳地利用参数（基于模型）和非参数（无模型或数据驱动）方法的优势。具体来说，半参数框架允许灵活的非参数模型来填补空白，并纠正参数模型中的低维模型误差。该框架采用了合奏的状态参数模型来表示预测或状态估计的不确定性，而全概率分布估计的非参数模型。在每个滤波或预测步骤中，通过从由非参数模型估计的模型误差分布中采样个体校正来更新集合。这些抽样修正将自动纠正模型中的偏差，并在必要时扩大不确定性，以符合现实。非参数模型的演化通常需要以参数模型的高维状态为条件，这是当前方法所不允许的。换句话说，信息必须在两个方向上流动：非参数模型校正参数模型，但也被参数模型的当前状态所告知。为了克服这一关键挑战，监督降维技术将与一种学习非同构空间之间映射的新方法相结合。这将允许根据学习到的参数状态投影对非参数状态估计进行贝叶斯更新。该研究包括一种新的高阶无迹集合预测，将形成高阶卡尔曼滤波器的基础。这些进步将最大限度地利用可用的计算资源，因为高阶集合预报和滤波方法可以在资源允许的情况下从小集合扩展到大集合。高阶方法将通过估计状态估计和预测的高阶矩来提高精度和不确定性量化。对于集合预报，将应用一种新的多元求积方法，该方法使用高阶矩的秩1张量分解作为求积节点。对于卡尔曼更新，将基于从Kushner方程（其完全描述了真实解）导出的矩方程的最大熵闭合，使用高阶方程。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，同时减轻模型误差和维数灾难。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.