How can we use mathematical optimization, uncertainty quantification, and asymptotic statistics to develop a computational framework that takes us from data and training statistical models to actually making robust, risk-averse decisions based on the predictions of these models?
Several techniques from artificial intelligence such as deep and reinforcement learning promise to provide the key to solving complex problems in engineering and the natural sciences. However, many of the resulting approaches are typically limited to simulation, not optimization or optimal control. Simulations are useful, but they alone do not provide us with optimal designs or control policies. This requires techniques of mathematical optimization. Since, the simulations are also subject to the underlying uncertainty in the data as well as the choice of model and the training procedure, need to understand and more importantly, quantify the uncertainty in the outputs. This allows us to use such statistical models as components in our decision-making problems. Finally, the dependence on data and uncertainty indicate that we need stability and statistical estimation procedures to properly evaluate the computable optimal solutions.
Theses in this project should seek to investigate the promises made by new techniques in scientific machine learning. The central themes should be based on the question: How can we make optimal decisions based on the predictions of novel, learning-based simulations of complex processes? There are many potential focus areas, but the underlying goal should be to provide a computational framework that takes us from data and modelling, beyond simulation, to optimization and optimal control.
The students will become familiar with a wide array of modern techniques from mathematical optimization, optimal control, optimization under uncertainty, and scientific machine learning. They will also gain valuable experience programming and solving complex mathematical problems.
Students should have some familiarity with the basics of nonlinear optimization, real analysis, programming in Python, Julia or C++, as well as some knowledge of probability and statistics. Ideally, the students will have taken courses in higher mathematics such as numerical analysis, functional analysis, and partial differential equations.
- Thomas Surowiec