CAREER: Learning Design Representations: The Effect of Differential Geometric Manifolds on the Inference of Design Structure

这个教师早期职业发展计划（CAREER）补助金将回答以下问题：“什么时候（或在什么条件下）将多个已知的微分几何流形合并到学习算法中显着提高学习设计表示的样本效率，给定数据？“找到设计产品和系统的有用数学表示是该领域长期存在的基础性开放研究问题之一。它几乎影响到每一个主要的设计任务，无论是优化，理解决策和偏好，制造计划，不确定性量化，还是研究设计认知，行为或创造力等等。但是找到有用的表示是困难的，因为它们必须同时描述多个数学结构，这些结构的类型在设计的形状、行为和功能上各不相同。如果成功的话，这种学习的设计表示可以：（1）加速技术和经济竞争力-例如，通过改进优化收敛-（2）使得能够对比目前可能的更复杂的系统进行不确定性量化-例如，在具有混沌动力学的系统中，如大涡模拟;以及（3）允许快速、准确的制造分析、计算机几何查询和设计合成-例如，通过为分析算法提供紧凑的搜索空间。通过将三个新的数学领域-微分几何，统计学和机器学习（ML）-与工程设计相结合，这项工作将推进如何推断复杂系统的基础数学表示背后的科学，例如，通过使现有的设计方法从单一的欧几里德表示扩展到一般的微分几何。与过去的设计表示方法不同，这种同时使用微分几何和ML的方法使研究能够回答有关有限样本中系统设计的实际限制的基本问题，当需要多种不同类型的表示时-生成的知识将回答其他科学领域中出现的相关问题。教育和推广将包括：（1）一套研究生暑期课程，以非传统的部分研讨会，部分暑期学校形式进行实验，以帮助将数学新领域的关键成果转化为工程设计;（2）开源，研究和教学基础设施，以扩大对设计和机器学习交叉领域主题的访问;以及（3）跨领域的教育和推广活动，重点是增加工业相关性和代表性不足的群体在学院和K-12级。该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准。

This Faculty Early Career Development Program (CAREER) grant will answer the following question: "When (or under what conditions) does incorporating and merging multiple, known differential geometric manifolds into a learning algorithm significantly improve the sample efficiency of learning a design representation, given data?" Finding useful mathematical representations of designed products and systems is one of the field's long-standing, fundamental, open research problems. It affects almost every major design task, whether that be optimization, understanding decision making and preferences, manufacturing planning, uncertainty quantification, or studying design cognition, behavior, or creativity, among others. But finding useful representations is difficult, because they must concurrently describe multiple mathematical structures, whose types vary across a design's shape, behavior, and function. If successful, such learned design representations can: (1) accelerate technological and economic competitiveness — e.g., by improving optimization convergence — (2) enable uncertainty quantification for more complex systems than presently possible — e.g., in systems with chaotic dynamics like Large Eddy Simulation; and (3) permit fast, accurate manufacturing analysis, computer geometry queries, and design synthesis — e.g., by providing compact search spaces for analysis algorithms. By combining advances across three novel mathematical areas -- Differential Geometry, Statistics, and Machine Learning (ML) — with Engineering Design, the work will advance the science behind how to infer underlying mathematical representations for complex systems, e.g., by enabling existing design methods to extend from single Euclidean representations to general Differential Geometry. Unlike past approaches to design representation, this approach of using both differential geometry and ML enables the research to answer fundamental questions regarding the practical limits of what is "learnable" about a system's design in finite samples when multiple, diverse types of representation are needed — the generated knowledge will answer related questions that arise across other scientific fields. Education and outreach will include: (1) a set of graduate summer programs that experiment with an unconventional part-workshop, part-summer school format for helping translate key results from new areas of mathematics into Engineering Design; (2) open-source, research and teaching infrastructure for broadening access to topics at the intersection of Design and Machine Learning; and (3) cross-cutting education and outreach activities that focus on increasing industrial relevance and participation of underrepresented groups in research and education at the college and K-12 level.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.