Enhanced Statistical Learning for Physical Systems Exploiting Non-Standard Constraints

现实世界的系统通常自然地受到物理规律或人类行为模式的约束，通过将这些约束合并到推理机制中，可以显著增强对此类系统的理解。然而，多重约束的存在可能会使统计学习过程复杂化。这个项目旨在开发新的统计方法来帮助解决现实世界中的问题，在这些问题中，许多复杂的约束构成了推理的挑战。动机直接来自三个具体的应用：i)从电子形状因子数据中提取质子半径，这是原子物理学中的一个基本问题，由于不同实验模式的结果之间的异常而具有很高的相关性；ii)描述风速与风力涡轮机产生的功率之间的关系，风力涡轮机是增长最快的可再生能源之一；以及iii)描述交通流模式与交通速度，这是交通工程研究的关键对象。该项目将结合机器学习和贝叶斯非参数学的想法，开发出在受限决策问题中进行统计上可靠和计算上有效的推理方法。该项目旨在为科学和工程应用中的推理开发一种原则性的概率方法，在这些应用中，各种物理约束提供了关于推理关键对象的先验知识。这样的对象可以对应于一条曲线或一组曲线或密度函数。PI和co-PI将开发新的统计方法，用于同时合并受实际科学和工程应用激励的多个形状约束，同时可广泛推广到所考虑的应用之外。重点是在一个灵活的非参数贝叶斯模型中得到各种约束的等价表示，并在这些约束空间上发展新的先验分布。贝叶斯方法对于获得不确定性估计很有吸引力，而PI和co-PI旨在为贝叶斯不确定性测量在当前环境下的频率有效性提供严格的理论保证。此外，PI和共同PI建议证明，包括约束可以减少不确定性，而不确定性将随后导致更好的科学结论。方法论的发展将伴随着高效的计算算法，以满足特定应用程序甚至更多应用程序的可扩展性要求。为了最大限度地发挥开发的方法的影响，PI和共同PI将与特定应用领域的专家密切合作。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Real world systems are often naturally constrained by physical laws or human behavioral patterns, and understanding of such systems can be significantly enhanced by incorporating said constraints into the inferential mechanism. However, the presence of multiple constraints can complicate the process of statistical learning. This project aims to develop novel statistical methods to help solve real world problems where a multitude of complex constraints pose inferential challenges. Motivations are drawn directly from three concrete applications: i) extracting the radius of proton from electric form-factor data, a fundamental problem in atomic physics which is of high relevance due to anomalies between results from different modes of experimentation, ii) describing the relationship between wind velocity and power derived from wind-turbines, which is one of the fastest growing renewable sources of energy and iii) describing the traffic flow pattern with traffic speed, a key object of research in traffic engineering. The project will bring together ideas from machine learning and Bayesian nonparametrics to develop statistically sound and computationally efficient methods of inference in constrained decision problems. The project aims to develop a principled probabilistic approach towards inference in scientific and engineering applications where various physical constraints provide a priori knowledge regarding key objects of inference. Such objects may correspond to a single curve or a collection of curves or density functions. The PI and co-PI will develop novel statistical methods for simultaneous incorporation of multiple shape constraints motivated by real scientific and engineering applications, while being broadly generalizable beyond the considered applications. Emphasis is laid on obtaining equivalent representations of various constraints within a flexible nonparametric Bayesian model, and developing novel prior distributions on these constrained spaces. The Bayesian approach is attractive to obtain uncertainty estimates, and the PI and co-PI aim to develop rigorous theoretical guarantees for the frequentist validity of Bayesian uncertainty measures in the present setting. In addition, the PI and co-PI propose to demonstrate that including the constraints diminishes the uncertainty which will subsequently lead to better scientific conclusions. The methodological developments will be accompanied by efficient computation algorithms that meet the scalability demanded by the specific applications and beyond. To maximize the impact of the methodology developed, the PI and co-PI will closely collaborate with domain experts for the specific applications.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.