From Structure Discovery to Scientific Machine Learning: Learning Operators in Dynamical Systems

This talk is a short tour of how I entered scientific machine learning through structure discovery in dynamical systems. I will begin with Sparse Identification of Lax Operators (SILO), a framework I developed for learning Lax pairs directly from data. SILO treats integrability as an operator learning problem, uses sparsity to expose the algebraic backbone of Hamiltonian systems, and serves as both a discovery tool and an integrability diagnostic. I will then describe how this line of thinking grew into TAILWINDs, a program for building reduced models in the coordinates that capture coherent structures that live on low-dimensional manifolds. These ideas connect symbolic regression, numerical analysis, and operator learning into a single workflow for understanding coherent structures and near-integrable behavior.

The overarching theme of the talk is how computational experimentation, geometry, and data can work together to reveal structure that is hard to see analytically. Along the way, I will show examples from PDEs, rigid-body dynamics, and nonlinear waves, and outline how these tools inform reduced-order modeling, scientific computation, and future work on structure-aware learning for complex physical systems.