Seminars and Colloquia by Series

Monte Carlo methods for the Hermitian eigenvaue problem

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 25, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE https://bluejeans.com/884917410
Speaker
Robert WebberCourant Institute

In quantum mechanics and the analysis of Markov processes, Monte Carlo methods are needed to identify low-lying eigenfunctions of dynamical generators. The standard Monte Carlo approaches for identifying eigenfunctions, however, can be inaccurate or slow to converge. What limits the efficiency of the currently available spectral estimation methods and what is needed to build more efficient methods for the future? Through numerical analysis and computational examples, we begin to answer these questions. We present the first-ever convergence proof and error bounds for the variational approach to conformational dynamics (VAC), the dominant method for estimating eigenfunctions used in biochemistry. Additionally, we analyze and optimize variational Monte Carlo (VMC), which combines Monte Carlo with neural networks to accurately identify low-lying eigenstates of quantum systems.

Time-parallel wave propagation in heterogeneous media aided by deep learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 23, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Richard TsaiUT Austin

 

We present a deep learning framework for learning multiscale wave propagation in heterogeneous media. The framework involves the construction of linear feed-forward networks (experts) that specialize in different media groups and a nonlinear "committee" network that gives an improved approximation of wave propagation in more complicated media.  The framework is then applied to stabilize the "parareal" schemes of Lions, Maday, and Turinici, which are time-parallelization schemes for evolutionary problems. 

Theoretical guarantees of machine learning methods for statistical sampling and PDEs in high dimensions

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 2, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Yulong LuUniversity of Massachusetts Amherst

Neural network-based machine learning methods, inlcuding the most notably deep learning have achieved extraordinary successes in numerious  fields. In spite of the rapid development of learning algorithms based on neural networks, their mathematical analysis are far from understood. In particular, it has been a big mystery that neural network-based machine learning methods work extremely well for solving high dimensional problems.

In this talk, I will demonstrate the power of  neural network methods for solving two classes of high dimensional problems: statistical sampling and PDEs. In the first part of the talk, I will present a universal approximation theorem of deep neural networks for representing high dimensional probability distributions. In the second part of the talk, I will discuss a generalization error bound of the Deep Ritz Method for solving high dimensional elliptic problems. For both problems,  our theoretical results show that neural networks-based methods  can overcome the curse of dimensionality.

A Few Thoughts on Deep Learning-Based Scientific Computing

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 26, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Haizhao YangPurdue University

The remarkable success of deep learning in computer science has evinced potentially great applications of deep learning in computational and applied mathematics. Understanding the mathematical principles of deep learning is crucial to validating and advancing deep learning-based scientific computing. We present a few thoughts on the theoretical foundation of this topic and our methodology for designing efficient solutions of high-dimensional and highly nonlinear partial differential equations, mainly focusing on the approximation and optimization of deep neural networks.

On the Continuum Between Models, Data-Driven Discovery and Machine Learning: Mapping the Continuum of Molecular Conformations Using Cryo-Electron Microscopy

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 19, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Roy Lederman Yale University

Cryo-Electron Microscopy (cryo-EM) is an imaging technology that is revolutionizing structural biology. Cryo-electron microscopes produce a large number of very noisy two-dimensional projection images of individual frozen molecules; unlike related methods, such as computed tomography (CT), the viewing direction of each particle image is unknown. The unknown directions, together with extreme levels of noise and additional technical factors, make the determination of the structure of molecules challenging. While other methods for structure determination, such as x-ray crystallography and nuclear magnetic resonance (NMR), measure ensembles of molecules, cryo-electron microscopes produce images of individual molecules. Therefore, cryo-EM could potentially be used to study mixtures of different conformations of molecules. Indeed, current algorithms have been very successful at analyzing homogeneous samples, and can recover some distinct conformations mixed in solutions, but, the determination of multiple conformations, and in particular, continuums of similar conformations (continuous heterogeneity), remains one of the open problems in cryo-EM. In practice, some of the key components in “molecular machines” are flexible and therefore appear as very blurry regions in 3-D reconstructions of macro-molecular structures that are otherwise stunning in resolution and detail.

We will discuss “hyper-molecules,” the mathematical formulation of heterogenous 3-D objects as higher dimensional objects, and the machinery that goes into recovering these “hyper-objects” from data. We will discuss some of the statistical and computational challenges, and how they are addressed by merging data-driven exploration, models and computational tools originally built for deep-learning.

This is joint work with Joakim Andén and Amit Singer.

Numerical methods for solving nonlinear PDEs from homotopy methods to machine learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 12, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Wenrui HaoPenn State University

Many systems of nonlinear PDEs are arising from engineering and biology and have attracted research scientists to study the multiple solution structure such as pattern formation. In this talk, I will present several methods to compute the multiple solutions of nonlinear PDEs. In specific, I will introduce the homotopy continuation technique to compute the multiple steady states of nonlinear differential equations and also to explore the relationship between the number of steady-states and parameters. Then I will also introduce a randomized Newton's method to solve the nonlinear system arising from neural network discretization of the nonlinear PDEs. Several benchmark problems will be used to illustrate these ideas.

Optimal-Transport Bayesian Sampling in the Era of Deep Learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, August 24, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Bluejeans (online) https://bluejeans.com/197711728
Speaker
Prof. Changyou ChenUniversity at Buffalo

Deep learning has achieved great success in recent years. One aspect overlooked by traditional deep-learning methods is uncertainty modeling, which can be very important in certain applications such as medical image classification and reinforcement learning. A standard way for uncertainty modeling is by adopting Bayesian inference. In this talk, I will share some of our recent work on scalable Bayesian inference by sampling, called optimal-transport sampling, motivated from the optimal-transport theory. Our framework formulates Bayesian sampling as optimizing a set of particles, overcoming some intrinsic issues of standard Bayesian sampling algorithms such as sampling efficiency and algorithm scalability. I will also describe how our sampling framework be applied to uncertainty and repulsive attention modeling in state-of-the-art natural-language-processing models.

https://bluejeans.com/197711728

Langevin dynamics with manifold structure: efficient solvers and predictions for conformational transitions in high dimensions

Series
Applied and Computational Mathematics Seminar
Time
Monday, June 22, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/963540401
Speaker
Dr. Yuan GaoDuke University

Please Note: virtual (online) seminar

We work on Langevin dynamics with collected dataset that distributed on a manifold M in a high dimensional Euclidean space. Through the diffusion map, we learn the reaction coordinates for N which is a manifold isometrically embedded into a low dimensional Euclidean space. This enables us to efficiently approximate the dynamics described by a Fokker-Planck equation on the manifold N. Based on this, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of N and enjoys the weighted l^2 convergence to the Fokker-Planck equation. The proposed upwind scheme leads to a Markov chain with transition probability between the nearest neighbor points, which enables us to directly conduct manifold-related computations such as finding the optimal coarse-grained network and the minimal energy path that represents chemical reactions or conformational changes. To acquire information about the equilibrium potential on manifold N, we apply a Gaussian Process regression algorithm to generate equilibrium potentials for a new physical system with new parameters. Combining with the proposed upwind scheme, we can calculate the trajectory of the Fokker-Planck equation on N based on the generated equilibrium potential. Finally, we develop an algorithm to pullback the trajectory to the original high dimensional space as a generative data for the new physical system. This is a joint work with Nan Wu and Jian-Guo Liu.

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