Ge Liu: Structured Generative Modeling for Scientific Data: Geometry, Symmetry, and Variable Length

Modern diffusion and flow-based generative models have become powerful tools for scientific discovery, yet many scientific problems go beyond probability paths defined in fixed-dimensional Euclidean spaces. Biological trajectories may lie on unknown manifolds, physical systems may obey Lie group symmetries, and protein or molecular design may require variable-length generation. This talk presents recent work from our group on generative modeling methods that respect these structures. First, I will present a flow-matching method for trajectory inference and generation on unknown manifolds. We learn a geodesically convex latent space that enables meaningful interpolation, and then pull the learned transport dynamics back to the data space. This allows the model to discover the geometry of a general scientific data manifold and use it for more effective transport and generation. Second, I will describe Trivialized Generative Models, a new family of generative models for Lie groups. TGM learns endpoint-constrained paths in a fixed Lie algebra and lifts them to the Lie group, providing a simple and powerful way to define valid group-valued generative dynamics. This endpoint formulation enables flexible path design beyond standard geodesic or exponential interpolation, avoids key complications of general Riemannian generative models, naturally supports non-compact groups, and extends to few-step consistency models and momentum-based higher-order dynamics. Finally, I will introduce Generalized Poisson Flow for variable-length protein design, where the model jointly learns the evolution of length and the generation of length-dependent multimodal features. Our method achieves state-of-the-art empirical performance across unconditional structure and sequence generation, motif scaffolding, and peptide co-design, validating a new paradigm for flexible-length protein design. Together, these works suggest a broader foundation for scientific generative modeling: designing probability paths that respect the geometry, symmetry, and variable-dimensional structure of scientific data.