UCSB Applied Math/PDE/Data Science Seminar

Abstract: The surface quasi-geostrophic (SQG) equation is a fundamental model in fluid dynamics that describes how temperature variations on the surface of a rotating fluid (like the atmosphere or ocean) can influence motion. In this talk, we study both the inviscid (no friction) and α-dissipative (with fractional diffusion) versions of this equation, collectively known as the α-SQG equations. To better understand these models, we first introduce a related system called the Double Dissipative SQG (DSQG) equations and show that solutions to this system behave nicely under certain conditions. We then investigate what happens as the effects of viscosity (or dissipation) become very small—a process known as taking the inviscid limit. Using energy estimates, we prove that solutions remain stable and converge as this limit is taken. Finally, we show that the limiting solution satisfies the weak form of the α-SQG equation, and with enough smoothness, it becomes a classical solution. Throughout, I’ll give an overview of the mathematical tools involved and explain how these results contribute to our broader understanding of fluid motion and turbulence models.

Host: Michael Gulas

Abstract: For a class of Schrodinger equations with locally analytic nonlinear terms, we treat the inverse problem of theoretically determining the potential and nonlinear interaction strength by measurements of the normal derivative on an arbitrary part of the boundary. In particular, we establish unique determination and stability of the coefficients with respect to the Neumann data provided a priori knowledge of the coefficients on some neighborhood of the boundary. This is joint work with my advisor, Hanming Zhou.

Host: Hanming Zhou

Abstract: (This is joint work with Karina Behera, Tron Omland, and Nicole Wu) Quantum metric spaces were originally developed by Marc Rieffel to address some statements in the particle physics literature, and they provide a notion of distance between quantum states, and therefore, a notion of distance between elements of an object in quantum information theory known as the density space. However, there already exist well-known metrics that provide distances between elements of a density space such as the Bures metric. Our work establishes comparisons of these metrics in a topological and metric sense. This work is partially supported by NSF grant DMS-2316892.

Host: Therese Landry

Abstract: This talk presents a control-oriented perspective on Scientific Machine Learning (SciML) for modeling, optimization, and control of dynamical systems. SciML provides a unifying computational paradigm that integrates physics-based models, optimization algorithms, and control policies within a differentiable programming framework. This synthesis enables computation of structured gradients for constrained system identification, learning-to-optimize, and learning-based control while preserving interpretability, stability, and physical consistency. Three recent advances will be highlighted. First, differentiable predictive control, a SciML approach that merges model predictive control with gradient-based learning to enable scalable, self-supervised training of explicit control policies suitable for real-time deployment on embedded hardware. Second, an operator-splitting formulation for neural differential-algebraic equations that integrates mechanistic dynamics with neural components to achieve robust extrapolation in systems with implicit constraints and conservation laws. Third, a self-supervised learning-to-optimize framework for mixed-integer nonlinear programs that provides feasibility guarantees and high-quality approximate solutions in milliseconds. Together, these advances demonstrate how SciML can unlock new capabilities for the modeling, optimization, and control of complex dynamical systems, with applications in power grid and building energy management.

Host: Paul J. Atzberger

Abstract: The quest to describe quantum states in a way that parallels classical probability theory goes back to the early days of quantum theory. Central to this effort has been the development of non-commutative versions of functional calculus and corresponding quasi-probability distributions, most notably, the Weyl functional calculus and Wigner distributions. A derivative classical probability distribution, known as the Husimi function, can be obtained from the Wigner distribution via Gaussian smearing in phase space. In this talk we explain the origin of the smearing in terms of continuous quantum measurements, lifting it from a convolution in phase space to a convolution in Hilbert space, via a novel notion of a Gaussian semigroup. This formalism yields a practical and canonical way to represent quantum states with probability distributions on Euclidean space for an arbitrary choice of self-adjoint observables.

Host: Therese Landry