Phase Space Foundations for Scientific Machine Learning

Objective

Science and engineering have long relied on interpretable, parsimonious models of signals and systems. Today there is a shift to highly-parameterized neural networks trained on massive datasets. This often makes it hard to understand what is taken from physics and what from data, and what mechanisms determine when the methods succeed or fail. There is a need for solid foundations for scalable and interpretable scientific machine learning. A key idea is to work in phase spaces inspired by Hamilton’s and Boltzmann’s formalisms. Phase spaces lift problems into higher-dimensional representations but in return render descriptions of a broad range of phenomena simple and interpretable. They reveal the universal local structure which is key for efficient computation, learning, and interpretability. In PhaseShift, we will: - Develop theory and designs for operators that leverage the universal Hamiltonian structure underlying applications from tomography to fluid flows, as described by microlocal analysis. We will prove that phase-space locality significantly improves sample efficiency and Lipschitz stability over existing networks. - Introduce a new generation of interpretable surrogate models built on discrete kinetic phase spaces, going well beyond PDE-based physics to phenomena like granular flows where classic PDEs do not suffice. We will also adapt kinetic theory principles to model general multivariate time series - Study of the role of scale interactions in both data and physics, and the related inductive biases for learning physics. In parallel, we will apply the developed tools to outstanding challenges in imaging and Earth science: end-to-end learning for cryoelectron tomography to visualize cells at near-atomic resolution, kinetic modeling of multistation seismic signals, and learning the dynamics of geophysical granular flows.