Research Interests

Exploring the intersection of PDEs, optimal control theory, and machine learning.

Professor Nadia Drenska

Research Interests

In my research, I use partial differential equations (PDEs), optimal control theory, and graph theory to analyze machine learning algorithms and investments, and give rigorous performance guarantees. I am a problem solver: I typically start with an application in mind, and learn and develop whatever methods I need in order to come up with a solution to the problem at hand. So far, I have bridged select ideas from viscosity solutions of PDEs, optimal control theory, and graph theory, to target problems in data science and investments. I have innovated and introduced continuous approaches to solve discrete problems from 'prediction with expert advice' and 'semi-supervised learning.' Moreover, my work proves that these approaches are asymptotically optimal.

A unifying theme for most of my research is adversarial two-player games, which arise in prediction problems as well as in p-Laplacian approaches to semi-supervised learning. My broader research goal is to use techniques from PDEs and ODEs to analyze and understand discrete problems from machine learning, graphs, and finance. Select applications are algorithm boosting, advertisement placing, stock investments, sports betting, self-driving car software, and data classification.

Another problem I have been working on is inferring body composition from 3D images. This project is joint with the Pennington Biomedical Research Center. You can find out more about it from the following article at SIAM Newsletter: Regression and Machine Learning Offer Insights About Body Composition Benchmarks

Publications

  • N. Drenska. Games on deBruijn Graphs and Cycle Means (submitted)
  • D. Mosaphir, J. Calder, and N. Drenska. Numerical Solution of a PDE Arising from Prediction with Expert Advice. (To appear in The European Journal of Applied Mathematics)
  • J. Calder and N. Drenska. Consistency of Semi-Supervised Learning, Stochastic Tug-of-War Games, and the p-Laplacian. Active Particles, Volume 4. 2024. https://doi.org/10.1007/978-3-031-73423-6_1
  • J. Calder and N. Drenska. Asymptotically Optimal Strategies for Online Prediction with History-Dependent Experts. Journal of Fourier Analysis and Applications, 2020 https://doi.org/10.1007/s00041-021-09815-4
  • N. Drenska and J. Calder. Online Prediction with History-Dependent Experts: The General Case. Communications on Pure and Applied Mathematics (CPAM), 2022, https://doi.org/10.1002/cpa.22049
  • N. Drenska and R. V. Kohn. A PDE Approach to the Prediction of a Binary Sequence with Advice from Two History-Dependent Experts. Communications on Pure and Applied Mathematics (CPAM), 2022, https://doi.org/10.1002/cpa.22071
  • N. Drenska and R.V. Kohn. Prediction with Expert Advice: a PDE Perspective. Journal of Nonlinear Science, 30(1):137-173, 2020, https://doi.org/10.1007/s00332-019-09570-3
  • N. Drenska. A PDE Approach to a Prediction Problem Involving Randomized Strategies. PhD thesis, New York University, 2017

Recent Talks

  • Body Composition: Insights through Regression and Machine Learning
    • LSU Math Club, November 2024
    • SIAM Annual Meeting, July 2024
  • Nadia Drenska's Machine Learning Journey
    • The Johns Hopkins University, April 2024
    • Louisiana State University, April 2024
  • Semi-Supervised Learning with the p-Laplacian in Geometric Methods in Machine Learning and Data Analysis
    • Numerical PDEs: Analysis, Algorithms, and Data Challenges, ICERM, March 2024
    • ICIAM, August 2023
  • Optimal Investment: Robo-Advising Under Small Changes of Risk Aversion
    • Joint Mathematics Meetings 2023
  • A PDE Interpretation of Prediction with Expert Advice
    • NJIT, January 2022
    • JMU Artificial Intelligence and Machine Learning Seminar Series, April 2021
    • Joint Mathematics Meetings, January 2021
    • OneWorld Machine Learning, December 2020
    • Optimal Control, Optimal Transport, and Data Science, IMA, November 2020
    • LMS Bath Symposium, August 2020
  • Johns Hopkins Applied Mathematics and Statistics Colloquium, October 2021
  • WPI Colloquium, January 2021
  • Two PDE Approaches to A Problem from Prediction with Expert Advice
    • IPAM, UCLA, May 2020
    • Analysis and Applied Mathematics Seminar, UIC, February 2020
  • PDE Approaches to Two Problems from Prediction with Expert Advice
    • Applied Interdisciplinary Mathematics Seminar, UMich 2019
  • A PDE Approach to Some Randomized-Strategy Two-Player Games
    • IMA Data Science Seminar, UMN 2018
    • Materials Working Group, NYU 2016
  • A PDE Approach to Prediction with Expert Advice
    • WPI STEM Faculty Launch, WPI, 2016
    • RPI Applied Math Days, RPI, 2016
    • SIAM Conference on Analysis of PDEs, Scottsdale AZ, 2015
      • Received SIAM Student Travel Award
    • Machine Learning Seminar, NYU, 2015
    • Materials Working Group, NYU, 2015

Online

This is a link to my adviser Robert V Kohn's presentation of joint work with Kangping Zhu and me at Waves, Spectral Theory, and Applications conference, 2015.

Awards

  • Moses Greenfield Prize for Outstanding Interdisciplinary Studies, Courant Institute, NYU, 2016
  • McCracken Scholarship, NYU
  • Rohn Truell Prize for Outstanding Undergraduate Student, DAM, Brown University, 2012
  • Sarah Dyer Barnes Scholarship, Brown University, 2011 - 2012
  • Sigma Xi, Brown University, 2012
  • Henry Parker Manning Prize Examination, First Prize, Brown University, 2011
  • First and Second Prizes on Math and Physics Olympiads, Bulgaria 2005-2007