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Oct 3, 2024: Kamyar Azizzadenesheli, Nvidia
Location: SMI 205 at 4pm
Oct 3, 2024: Kamyar Azizzadenesheli, Nvidia
Location: SMI 205 at 4pm
Title: AI+Science: Neural Operators for Accelerating Scientific Simulations and Design
Abstract: The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.
Oct 8 , 2024: Patrick Guidotti, UCI
Oct 8 , 2024: Patrick Guidotti, UCI
Location and time: TBA at 4pm
Title: Connecting the Dots
Abstract: We revisit the well-known kernel method of interpolation and, by taking a slightly unusual point view, show how it can be used (and modified in a natural way) for the purpose of gaining insight into the (geometric) structure of scattered data points such as point clouds. One of the advantages of the method is that it is global (albeit localizable) and does not require any direct explicit understanding of data points neighborhoods.
Oct 10, 2024: Boeing Colloquium - Eric vanden Eijnden (NYU)
Oct 10, 2024: Boeing Colloquium - Eric vanden Eijnden (NYU)
Location and time: SMI 205 at 4pm
Title: Generative modeling with flows and diffusions
Abstract: Generative models based on dynamical transport have recently led to significant advances in unsupervised learning. At mathematical level, these models are primarily designed around the construction of a map between two probability distributions that transform samples from the first into samples from the second. While these methods were first introduced in the context of image generation, they have found a wide range of applications, including in scientific computing where they offer interesting ways to reconsider complex problems once thought intractable because of the curse of dimensionality. In this talk, I will discuss the mathematical underpinning of generative models based on flows and diffusions, and show how a better understanding of their inner workings can help improve their design. These results indicate how to structure the transport to best reach complex target distributions while maintaining computational efficiency, both at learning and sampling stages. I will also discuss applications of generative AI in scientific computing, in particular in the context of Monte Carlo sampling, with applications to the statistical mechanics and Bayesian inference, as well as probabilistic forecasting, with application to fluid dynamics and atmosphere/ocean science.
Oct 15, 2024: Caroline Moosmuller, UNC
Oct 15, 2024: Caroline Moosmuller, UNC
Location: SMI 205 at 4pm
Title: Learning in the space of probability measures
Abstract: Many datasets in modern applications - from cell gene expression and images to shapes and text documents - are naturally interpreted as probability measures, distributions, histograms, or point clouds. This perspective motivates the development of learning algorithms that operate directly in the space of probability measures. However, this space presents unique challenges: it is nonlinear and infinite-dimensional. Fortunately, it possesses a natural Riemannian-type geometry which enables meaningful learning algorithms.
This talk will introduce the fundamentals of the space of probability measures and explore approaches to unsupervised, supervised, and manifold learning within this framework. We will discuss two key ideas: (1) a linearization approach, and (2) a "curved" approach utilizing finite-dimensional Riemannian submanifolds. Additionally, we will examine time evolutions on this space, including flows involving stochastic gradient descent and trajectory inference, with applications to analyzing the evolution of gene expression in single cells.
Oct 17, 2024: Deniz Bilman, University of Cincinnati
Oct 17, 2024: Deniz Bilman, University of Cincinnati
Location: SMI 205 at 4pm
Title: General rogue waves of infinite order
Abstract: We will present results from a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of this family of solutions: they are all in $L^2(\mathbb{R})$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this talk, we will define these solutions, establish their basic exact properties and asymptotic behavior, and describe tools for computing them accurately. This is joint work with Peter D. Miller.
Oct 22, 2024: Mateo Diaz, Johns Hopkins University
Oct 22, 2024: Mateo Diaz, Johns Hopkins University
Location: Condon Hall 135 at 2pm
Title: The radius of statistical efficiency
Abstract: Classical results in asymptotic statistics show that the Fisher information matrix controls the difficulty of estimating a statistical model from observed data. In this work, we introduce a companion measure of robustness of an estimation problem: the radius of statistical efficiency (RSE) is the size of the smallest perturbation to the problem data that renders the Fisher information matrix singular. We compute the RSE up to numerical constants for a variety of test bed problems, including principal component analysis, generalized linear models, phase retrieval, bilinear sensing, and matrix completion. In all cases, the RSE quantifies the compatibility between the covariance of the population data and the latent model parameter. Interestingly, we observe a precise reciprocal relationship between the RSE and the intrinsic complexity/sensitivity of the problem instance, paralleling the classical Eckart–Young theorem in numerical analysis.
Oct 24, 2024: Boeing Colloquium - Carola-Bibiane Schönlieb, Cambridge
Oct 24, 2024: Boeing Colloquium - Carola-Bibiane Schönlieb, Cambridge
Location: SMI 205 at 4pm
Title: Mathematical imaging: From geometric PDEs and variational modelling to deep learning for images
Abstract: Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of nonlinear partial differential equations, inverse problems, harmonic, stochastic and statistical analysis, and optimisation.
In this talk we will learn about some of these mathematical problems, about variational models and PDEs for image analysis and inverse imaging problems as well as recent advances where such mathematical models are complemented by deep neural networks.
The talk is furnished with applications to art restoration, forest conservation and cancer research.
Oct 31, 2024: Panos Stinis, PNNL
Oct 31, 2024: Panos Stinis, PNNL
Location: SMI 205 at 4pm
Title: When big neural networks are not enough: physics, multifidelity and kernels
Abstract: Modern machine learning has shown remarkable promise in multiple applications. However, brute force use of neural networks, even when they have huge numbers of trainable parameters, can fail to provide highly accurate predictions for problems in the physical sciences. We present a collection of ideas about how enforcing physics, exploiting multifidelity knowledge and the kernel representation of neural networks can lead to significant increase in efficiency and/or accuracy. Various examples are used to illustrate the ideas.
Nov 7, 2024: Boeing Colloquium - Xiaoye Sherry Li, Lawrence Berkeley National Laboratory
Nov 7, 2024: Boeing Colloquium - Xiaoye Sherry Li, Lawrence Berkeley National Laboratory
Location: SMI 205 at 4pm
Title: Interplay of Linear Algebra, Machine Learning, and High Performance Computing
Abstract: In recent years, we have seen a large body of research using hierarchical matrix algebra to construct low complexity linear solvers and preconditioners. Not only can these fast solvers significantly accelerate the speed of large scale PDE based simulations, but also they can speed up many AI and machine learning algorithms which are often matrix-computation-bound. On the other hand, statistical and machine learning methods can be used to help select best solvers or solvers' configurations for specific problems and computer platforms. In all these fields, high performance computing becomes an indispensable cross-cutting tool for achieving real-time solutions for big data problems. In this talk, we will show our recent developments in the intersection of these areas.
Nov 14 , 2024: Alina Chertok, NC State
Nov 14 , 2024: Alina Chertok, NC State
Location: SMI 205 at 4pm
Title: Navigating Uncertainty: Stochastic Methods for Nonlinear Systems
Abstract: Many important scientific problems involve several sources of uncertainties, such as model parameters and initial and boundary conditions. Quantifying these uncertainties is essential for many applications since it helps to conduct sensitivity analysis and provides guidance for improving the models. The design of reliable numerical methods for models with uncertainties has seen a lot of activity lately. One of the most popular methods is Monte Carlo-type simulations, which are generally good but inefficient due to the large number of realizations required. In addition to Monte Carlo methods, a widely used approach for solving partial differential equations with uncertainties is the generalized polynomial chaos (gPC), where stochastic processes are represented in terms of orthogonal polynomials series of random variables. It is well-known that gPC-based, spectral-type methods exhibit fast convergence when the solution depends smoothly on random parameters. However, their application to nonlinear systems of conservation/balance laws still encounters some significant difficulties. The latter is related to the presence of discontinuities that may develop in numerical solutions in finite time, triggering the appearance of aliasing errors and Gibbs-type phenomena. This talk will provide an overview of numerical methods for models with uncertainties and explore strategies to address the challenges encountered when applying these methods to nonlinear hyperbolic systems of conservation and balance laws.
Nov 21, 2024: Fatemeh Pourahmadian, UC Boulder
Nov 21, 2024: Fatemeh Pourahmadian, UC Boulder
Location: SMI 205 at 4pm
Title: Data-driven approaches to imaging and characterization of advanced materials from laser ultrasonic test data
Abstract: The first part of this talk highlights recent advances in the theory of inverse scattering germane to a new class of imaging functionals that enable spatiotemporal tracking of engineered (or manmade) processes in complex and/or unknown environments. More specifically, I will introduce the theory of differential imaging that is rooted in the factorization and linear sampling methods. The second part of the talk showcases an application of the sampling-based imaging indicators to laser ultrasonic imaging and characterization of additively manufactured (AM) components from limited-aperture test data. This includes a discussion of our recent efforts to (a) enhance and accelerate the waveform inversion process via deep learning in order to enable real-time remote sensing of AM processes, and (b) address challenges related to reconstructions from noisy measurements. The indicator maps from the sampling type imaging functionals are compared to those furnished by the state-of-the-art approaches to laser ultrasonic imaging. Our preliminary results highlight the unique advantages of ML-accelerated waveform tomography when applied to multi-fidelity experimental data.
Dec 5, 2024: Donsub Rim, Washington University in St. Louis
Dec 5, 2024: Donsub Rim, Washington University in St. Louis
Location: SMI 205 at 4pm
Title: A Low Rank Neural Representation of Nonlinear Shock Waves
Abstract: Despite decades of significant progress in scientific computing, real-time solvers of parametrized partial differential equations (pPDEs) describing nonlinear wave phenomena generally remains out of reach. One important obstacle is the apparent lack of low-dimensional structure in wave phenomena. For example, it is known that the Kolmogorov width decays slowly for solution manifolds of nonlinear conservation laws, prohibiting construction of efficient reduced basis methods. In this talk, I will discuss theoretical and computational results that point towards a new low-dimensional structure in convective problems, as described by a neural network architecture we call low rank neural representation (LRNR), that can potentially lead to new dimensionality reduction tools as well as to new real-time solvers applicable to various pPDEs. In particular, I will discuss how (1) a theoretical LRNR construction with small parameter dimension can efficiently approximate solutions to scalar nonlinear conservation laws involving arbitrarily complicated shock interactions, (2) LRNRs can be used within the physics informed neural networks (PINNs) to efficiently compute pPDE solutions.
This is talk is based on joint works with Woojin Cho (Yonsei U.), Kookjin Lee (Arizona State U.), Noseong Park (KAIST), Gerrit Welper (U. Central Florida), and Randall J. LeVeque (U. Washington).
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