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January 11, 2024: Cole Graham, Brown University
Location: Raitt 121 at 4pm
January 11, 2024: Cole Graham, Brown University
Location: Raitt 121 at 4pm
Title: Invasion in several guises
Abstract: The world teems with examples of invasion, in which one
steady state spatially invades another. Invasion can even display a
universal character: fine details recur in seemingly unrelated
systems. Reaction-diffusion equations provide a mathematical framework
for these phenomena. In this talk, I will discuss recent examples of
robust invasion patterns in reaction-diffusion equations, with an
emphasis on multiple dimensions.
January 18, 2024: Siting Liu, UCLA
January 18, 2024: Siting Liu, UCLA
Location and time: Raitt 121 at 4pm
Title: Enhancing PDE Computations and Score-Based Generative Models through Optimization
Abstract: This presentation explores optimization strategies for improving both partial differential equation (PDE) computations and score-based generative models (SGM). In the realm of numerical computations, we introduce a saddle point framework that capitalizes on the inherent structure of PDEs. Integrated seamlessly with existing discretization schemes, this framework eliminates the necessity for nonlinear inversions, paving the way for efficient parallelization. Shifting our focus to SGM, we delve into the mathematical foundations of the Wasserstein proximal operator (WPO). Specifically, we express it as the Wasserstein proximal operator of cross-entropy. By leveraging the PDE formulation of WPO, we propose a WPO-informed score model that demonstrates accelerated training and reduced data requirements.
January 23, 2024: Brendon G. Anderson, UC Berkeley
January 23, 2024: Brendon G. Anderson, UC Berkeley
Location and time: Raitt 121 at 4:00 pm
Title: Provably Robust Machine Learning through Structure-Aware Computation
Abstract: Standard machine learning (ML) algorithms exhibit catastrophic failures when subject to uncertainties in their input data, such as attacks generated by an adversary. Robustness against such uncertainties must be guaranteed in order to reliably deploy ML in safety-critical settings, such as aviation, autonomous driving, and healthcare. This talk presents recent theoretical and computational advancements in provably robust machine learning. We both introduce novel ML models endowed with mathematical proof of robustness, as well as optimization methods to certify the robustness of prior models. By exploiting key structures in the underlying certification problems, the proposed methods achieve state-of-the-art robustness and efficiency.
January 25, 2024: Montie S. Avery, Boston University
January 25, 2024: Montie S. Avery, Boston University
Location: Raitt 121 at 4pm
Title: Universal spreading into unstable states
Abstract: The emergence of complex spatial structures in physical systems often occurs after a simpler background state becomes unstable. Localized fluctuations then grow and spread into the unstable state, forming an invasion front which propagates with a fixed speed and selects a new stable state in its wake. The mathematical study of these invasion processes has historically been limited to systems with restrictive monotonicity properties (in PDE terms, a comparison principle). Such systems, however, inherently cannot describe the formation of complex spatiotemporal patterns, which is of particular interest both in nature and in manufacturing applications. On the other hand, formal calculations in the physics literature have long outlined a universal approach for predicting invasion speeds and associated selected states, valid for systems which do not obey comparison principles and instead exhibit complex spatiotemporal dynamics. This prediction scheme is often referred to as the marginal stability conjecture. In this talk, I will discuss the first proof of the marginal stability conjecture and explore how this can be used to make concrete predictions for physical systems.
January 30, 2024: Nicholas M. Boffi, NYU
January 30, 2024: Nicholas M. Boffi, NYU
Location: Raitt 121 at 4pm
Title: On flows and diffusions: from the many-body Fokker-Planck equation to stochastic interpolants
Abstract: Given a stochastic differential equation, its corresponding Fokker-Planck equation is generically intractable to solve because its high dimensionality prohibits the application of standard numerical techniques. In this talk, I will exploit an analogy between the Fokker-Planck equation and modern generative models from machine learning to develop an algorithm for its solution in high dimension. The method enables the computation of previously intractable quantities of interest, such as the entropy production rate of active matter systems, which quantifies the magnitude of nonequilibrium effects. I will then highlight how insight from the Fokker-Planck equation facilitates the development of a new class of generative models known as stochastic interpolants, which generalize state of the art diffusion models in several key ways that can be leveraged to improve practical performance. Along the way, I will argue that methods from machine learning offer a compelling solution for many fascinating high-dimensional mathematical problems that are currently out of reach with more traditional computational tools.
February 1, 2024: Rima Alaifari, ETH Zurich
February 1, 2024: Rima Alaifari, ETH Zurich
Location: Raitt 121 at 4pm
Title: “Stable or not: Robustness in imaging and scientific machine learning”
Abstract: Stability is crucial in applications that require deriving solutions from some input data. Classically, the notion of stability describes robustness under small perturbations of the input and regularization is employed to derive solutions from problems that lack this stable dependence.
In this talk, we visit different problems and methods that lack stability in some, probably less classical, sense. First, we discuss a phase retrieval problem which is not uniformly stable. This non-linear inverse problem cannot be tackled by classical regularization and we highlight possible connections between uniqueness and stability of this problem. Next, we take a look at robustness through the lens of adversarial attacks both for image classification and image reconstruction. While it is known that successful attacks can be designed for data-driven methods, we find that also classical regularization methods can be adversarially attacked.
The last part of the talk is devoted to operator learning and its stability with respect to discretizations. We propose a novel concept of neural operators that by-passes aliasing. These Representation equivalent Neural Operators (ReNOs) establish a unique and stable link between operators on infinite-dimensional spaces and their discrete realizations.
February 8, 2024: Eric Brechner, UW
February 8, 2024: Eric Brechner, UW
Location: Raitt 121 at 4pm
Title: N-dimensional solid modeling without topology
Abstract: Modern design and manufacturing of everything from toys to commercial airliners depends on solid modeling. Boolean operations on solid models are essential for their construction and deployment. The boundary representations of solids used by nearly all solid modeling systems rely on topological information to stitch together the numerically challenging surface intersections inherent in Boolean operations. This reliance on topology adds complexity to Boolean operations, leads to inevitable inconsistencies between geometry and topology, limits surface variation, and does not scale well beyond three dimensions. This talk presents a simple new approach to Boolean operations on solids that does not rely on topology. The algorithm supports arbitrary manifolds with unit normals in any dimension. Examples developed using the “BSpy” open source B-Spline Python package will be demonstrated.
February 15, 2024: Christina Graf, UBC
February 15, 2024: Christina Graf, UBC
Location: Raitt 121 at 4pm
Title: High Fidelity MRI by Time Optimal Control
Abstract: Magnetic Resonance Imaging and Spectroscopy provide detailed images and information on the chemical composition of the human body combining a strong, static magnetic field B0, and a radio frequency (RF) field B1. Together, they act on the net magnetization vector through the Bloch equations, which constitute a system of partial differential equations.
Imperfect B0 and B1 fields impact the temporal evolution of the magnetization vector, influencing the entire MR measurement. Dedicated RF pulses can alleviate the effects of these imperfect fields. One example is the class of adiabatic RF pulses, but they come with limitations, including high RF amplitude, pulse duration, and adaptability.
To overcome these limitations, RF pulse design through optimal control has shown excellent results in terms of flexibility. In this approach, robustness to field inhomogeneities is the primary objective within the cost functional, with the Bloch equations serving as constraints that describe the underlying physics. This method results in an excellent efficiency of magnetization performance.
Using two examples, namely magnetization inversion for in vivo MRI, and robust excitation for 31P MRS, the outstanding characteristics of RF pulse design by optimal control are demonstrated.
February 29, 2024: Deanna Needell, UCLA
February 29, 2024: Deanna Needell, UCLA
Location: Raitt 121 at 4pm
Title: Towards Transparency, Fairness, and Efficiency in Machine Learning
Abstract: In this talk, we will address several areas of recent work centered around the themes of transparency and fairness in machine learning as well as practical efficiency for methods with high dimensional data. We will discuss recent results involving linear algebraic tools for learning, such as methods in non-negative matrix factorization. We will showcase our derived theoretical guarantees as well as practical applications of those approaches. These methods allow for natural transparency and human interpretability while still offering strong performance. Then, we will discuss new directions in fairness including an example in large-scale optimization that allows for population subgroups to have better predictors than when treated within the population as a whole. We will conclude with work on compression and reconstruction of large-scale tensorial data from practical measurement schemes. Throughout the talk, we will include example applications from collaborations with community partners.
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