Interfacial problems in data science and PDEs

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.