This PhD project investigates how to incorporate symmetries, constraints, and topology into generative models without sacrificing expressiveness. The goal is to develop structure-preserving generative methods that maintain key properties of the data manifold such as neighborhood relations, connectivity, and meaningful clusters and transitions while enabling high-quality sampling and optimization. The project will explore ways to represent and measure these properties using tools from geometry and topology, and to incorporate these signals into training objectives and model design. We will also focus on the application of the above mentioned methods into the use case of small molecules and peptide design. The key question to be investigated here is how one can implement these concepts in adaptive design loops that translate in silico hypotheses into in-vitro evidence. In parallel, the project will use tools inspired by statistical mechanics to characterize learning dynamics in neural networks e.g., changes in representation geometry, and potential phase-transition-like behavior providing mechanistic insight into phenomena often treated as black-box.
