CRII: OAC: A Multi-fidelity Computational Framework for Discovering Governing Equations Under Uncertainty

了解物理系统的本质一直是人类的一项基本好奇心，使人们能够发现许多支配这些系统和过程的基本物理定律。通常，在这个过程中，人们发现更直接的解释更受青睐--体现在著名的奥卡姆剃刀或节俭原则中。带着类似的目标，这个项目进行基础研究，寻求在理解支配复杂物理系统在不确定条件下行为的物理学简约原理方面推进最先进的技术。由此产生的软件框架使用户能够利用来自广泛的物理系统的数据，通过分析所识别的数学方程来解锁其行为的重要方面。在这个框架下，项目调查员为研究人员和教育工作者社区提供了一个强大而顺从的工具，以理解和预测各种真实世界系统在存在不确定性的情况下的实际行为。该框架利用已有的理解具有相似行为的简化物理系统的知识，利用可解释的深度学习来发现简约的控制方程来描述复杂物理系统在不确定条件下的行为。还探索了利用低保真模型定义系统行为的新方法，提供了更高效的计算工具来结合先验知识。这些改进和开发的软件允许该框架应用于各种应用领域。该框架通过试验台问题进行了评估，例如20层建筑的行为和基准湍流建模问题。此外，由此产生的框架还通过合作应用于其他复杂性质的关键问题，例如野地火灾的蔓延、酶催化反应和污染物羽流的蔓延。该项目还强调，必须通过教育和外联活动确定简约的控制方程式，以描述复杂系统的行为。该项目的成果被提交给有声誉的期刊，并在国内和国际会议上发表。一名研究生作为项目的一部分接受指导，研究结果被纳入本科生的概率与机器学习课程。由项目调查员领导的K-12推广工作包括发现方程、机器学习基础知识的模块，以及使用软件根据在初中/高中暑期研讨会计划中展示的该项目的研究结果来推断数据。该奖项反映了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。

Understanding the nature of physical systems has always been an essential curiosity of humankind, enabling the discovery of many fundamental physical laws governing these systems and processes. Often, in the process, it was seen that a more straightforward explanation is preferred -- embodied in the famous Occam's razor or principle of parsimony. With a similar goal, this project conducts fundamental research seeking to advance the state-of-the-art in understanding the parsimonious principles of physics governing the behavior of a complex physical system under uncertainty. The resulting software framework enables the users to utilize data from a wide range of physical systems to unlock the important aspects of their behavior by analyzing the identified mathematical equations. With this framework, the project investigator provides the community of researchers and educators with a powerful and amenable tool to comprehend and predict the actual behavior of various real-world systems in the presence of uncertainty. Harnessing already developed knowledge in understanding simplified physical systems with similar behavior, the framework builds a novel paradigm using interpretable deep learning for discovering parsimonious governing equations to describe the behavior of complex physical systems under uncertainty. New approaches to utilizing low-fidelity models in defining the system's behavior are also explored, providing even more computationally efficient tools to incorporate prior knowledge. These advancements and the developed software allow the framework to be applied to various application domains. The framework is evaluated with testbed problems such as the behavior of a 20-story building and a benchmark turbulence modeling problem. Furthermore, the resulting framework is applied to other critical problems of complex nature through collaborations, e.g., the spread of wildland fires, enzyme-catalyzed reactions, and the spread of pollutant plumes. The project also emphasizes the importance of identifying parsimonious governing equations to describe the behavior of complex systems through education and outreach activities. Results from the project are submitted to reputable journals and presented at national and international conferences. One graduate student is mentored as part of the project, and findings are incorporated into an undergraduate probability and machine learning course. K-12 outreach efforts led by the project investigator include modules on the discovery of equations, machine learning basics, and using software to infer from data based on the research outputs from this project presented at middle/high school summer workshop programs.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.