Interfacial problems in data science and PDEs

该项目涉及数学模型的开发和分析，其中不同阶段或类之间的接口结构是一系列应用程序中出现的基本特征。在这个项目中专门解决的框架是肿瘤生长模型和机器学习分类系统对恶意攻击者的鲁棒性，后者在文献中也被称为对抗性攻击。在肿瘤生长的情况下，关键的界面是区分癌变区域和健康细胞区域的边界。跟踪和预测这一边界的形状对疾病的治疗和预后具有重要意义。鲁棒机器学习分类的关键接口是不同类别之间的决策边界。如果分类器具有非常不规则的决策边界，那么对手可以通过稍微干扰输入数据来产生错误的结果。这些攻击在计算机视觉中尤其引人注目，在计算机视觉中，人类视觉无法察觉的像素变化可以以高置信度欺骗训练有素的模型。该项目将通过研究这些界面的理论特性，开发新的计算算法和软件来模拟这些界面，从而提高我们在这两个应用领域的知识。这个项目也将涉及研究生的培训。在肿瘤生长的情况下，该项目将侧重于基于偏微分方程的细胞生长的流体力学模型。在理论方面，将建立这些模型的适定性，并理解模型选择和参数制度如何影响肿瘤边界的规律性。此外，将开发有效的数值求解器，特别关注对当前理论理解来说过于具有挑战性的模型实例，包括甚至不知道弱解的全局实时存在性的模型。在数据分类的情况下，将建立对对抗性学习对类接口的正则化效果的更好理解，这些见解将用于指导参数选择。将对抗性学习与关于最优运输距离的重心问题联系起来的理论工作也将进一步发展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project involves the development and analysis of mathematical models where the structure of interfaces between different phases or classes is an essential feature that arises in a range of applications. The frameworks specifically addressed in this project are models for tumor growth and the robustness of machine learning classification systems to malicious attackers, the latter also known in the literature as adversarial attacks. In the case of tumor growth, the key interface is the boundary separating the cancerous region and the healthy cell region. Tracking and predicting the shape of this boundary has important implications for disease treatment and prognosis. The key interfaces for robust machine learning classification are the decision boundaries between different classes. If a classifier has extremely irregular decision boundaries, then an adversary can produce erroneous results by slightly perturbing the input data. These attacks are particularly striking in computer vision, where pixel changes that are imperceptible to human vision can fool well-trained models with high confidence. This project will advance our knowledge in both of these applications by studying the theoretical properties of these interfaces and by developing new computational algorithms and software to simulate these interfaces. This project will also involve the training of graduate students. In the case of tumor growth, the project will focus on fluid mechanical models for cell growth based on partial differential equations. On the theoretical side, the well-posedness of these models and an understanding of how model choice and parameter regimes affect the regularity of the tumor boundary will be established. In addition, efficient numerical solvers will be developed, paying particular attention to model instances that are too challenging for the current theoretical understanding, including models where even the global-in-time existence of weak solutions is unknown. In the case of data classification, a better understanding of the regularizing effect that adversarial learning has on class interfaces will be established, and these insights will be used to guide parameter choices. Theoretical work that has connected adversarial learning to barycenter problems with respect to optimal transport distances will also be further developed.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.