Organizing a minisymposium: Deep Numerical Analysis and Optimization for PDEs, 2020 International Conference of Spectral and High Order Methods
Machine learning approaches for solving nonlinear or high-dimensional partial differential equations (PDEs) and inverse problems have been an important direction in applied and computational mathematics. Especially, the “unreasonable effectiveness” of deep learning for massive datasets accumulated in research activities posed numerous mathematical and algorithmic challenges along the path towards gaining deeper understandings of machine learning in scientific computing. Deep learning has been shown to include various wavelets, adaptive finite element methods, and multigrid methods. This mini-symposium covers a broad range of related topics including new developments of neural networks, fast training algorithms, theoretical foundation of deep learning, and relevant applications aiming at efficient solvers for nonlinear or high-dimensional PDEs and inverse problems.
Organizing a minisymposium: Machine learning for interatomic potentials, 2020 SIAM Conference on Mathematical Aspects of Materials Science
A promising new way to create interatomic potentials for molecular dynamics simulations is to take energies and forces from electronic structure calculations and fit a carefully crafted functional ansatz to this reference data. Determining such potentials is a challenging high-dimensional function approximation problem whose effective solution requires a combination of mathematical analysis and physical insight. This minisymposium will bring together physicists, chemists and mathematicians to provide an overview of the current state of the art and explore future directions.
Organizing a minisymposium: Mathematical Understanding and Applications of Learning with Networks, 2020 SIAM Conference on Mathematics of Data Science
Machine learning together with massive datasets has shown tremendous success in a variety of challenging applications. Yet, such a triumph in learning methodologies still lacks rigorous mathematical foundation in various problems. Over the past few years, much progress has been made in developing mathematical theories and computational tools from different perspectives, including approximation theory, optimization theory, geometry theory, and graph theory with applications in social science, biological science, as well as physical science. This minisymposium brings various researchers and experts in these areas together, particularly with the aim of high dimensional data analysis (computation) with neural networks or complex networks; the talks reflect the collaborative, multifaceted nature of the mathematical theory and applications of machine learning.
Organizing a minisymposium: Theory and Algorithms for Data Science, the 2020 AMS Spring Sectional Meeting at Purdue University
Motivated by the extremely large amounts of data streaming from scientific research and daily life, recent years have witnessed exciting developments in the theory and practice of data science for obtaining instantaneous insight from massive datasets. This minisymposium aims at bringing together mathematicians, statisticians, and computer scientists interested in the theoretical aspects of data science, with diverse background and expertise ranging from approximation theory, optimization methods, generalization performance, and statistical apporaches to modeling high-dimensional and nonlinear systems; the talks reflect the collaborative, multifaceted nature of the mathematical theory and applications in interdisciplinary domains.
Organizing a minisymposium: Machine Learning for Solving Partial Differential Equations and Inverse Problems, the 2nd Annual Meeting of the SIAM Texas-Louisiana Section
Machine learning (ML) approaches have gained increasing attention as a potential tool for effective solutions of nonlinear or high-dimensional partial differential equations (PDEs) and inverse problems, especially those with uncertainty and stochastic effects. A deeper understanding of the “unreasonable effectiveness” of deep learning will be important to address various mathematical and algorithmic challenges in ML for scientific computing. This mini-symposium covers a broad range of related topics including new developments in physics-informed neural networks, learning for high frequency wave propagations in random media, regularization techniques based on deep learning, theoretical foundation of deep learning, with a focus on efficient solvers for nonlinear or high-dimensional PDEs and inverse problems.
Organizing Workshop on High-Dimensional Learning and Computation in Physics, Department of Mathematics, National University of Singapore, June 2019
This workshop gathers young researchers to discuss recent progress in high-dimensional learning and computation techniques bridging recent advances in machine learning and computational physics. The goal is to promote stronger collaborations among active researchers in these frontiers of scientific computing and applied mathematics.
The “unreasonable effectiveness” of deep learning for massive datasets posed numerous mathematical and algorithmic challenges along the path towards gaining deeper understandings of new phenomena in machine learning. This minisymposium aims at bringing together applied mathematicians interested in the mathematical aspects of deep learning, with diverse background and expertise ranging from approximation theory, optimization methods, and generalization performance to modeling high-dimensional scientific computing problems and nonlinear physical systems; the talks reflect the collaborative, multifaceted nature of the mathematical theory and applications of deep neural networks. Section 1 concerns the approximation capacity and optimization of deep learning, Section 2 concerns the generalization and perturbation error of deep learning, Section 3 concerns the applications of deep learning.
Integral methods are useful tools in applied science and engineering. In particular, they are an important topic for large-scale scientific computing. Many challenges remain open and attract much attention especially in the high-frequency regime. This minisymposium focuses on recent advances in integral equations and integral transforms for highly oscillatory phenomena, including new formulations for high-frequency wave propagation, efficient and accurate discretizations, novel fast algorithms and their implementation based on locally rank-structured matrices and non-oscillatory phase functions, with applications in various imaging science and computational electromagnetism. Section 1, Section 2.
Many objects of interest in imaging science exhibit a low-dimensional structure, which could mean, for instance, low sparsity of a vector, low-rank property of a large matrix, or low-dimensional manifold model for a data set. Many successful methods rely on deep understanding and clever exploitation of such low-dimensional structures. The goal of this mini-symposium is to bring together researchers actively working on imaging techniques based on low-dimensional models, and to explore some recent state-of-the-art work in scientific computation, machine learning and optimization related with imaging science. Section 1, Section 2, Section 3.
Eigenvalue problem is the essential part and the computationally intensive part in many applications in a variety of areas, including, electron structure calculation, dynamic systems, machine learning, etc. In all these areas, efficient algorithms for solving large-scale eigenvalue problems are demanding. Recently many novel scalable eigensolvers were developed to meet this demand. The choice of an eigensolver highly depends on the properties and structure of the application. This minisymposium in- vites eigensolver developers to discuss the applicability and performance of their new solvers. The ultimate goal is to assist computational specialists with the proper choice of eigensolvers for their applications. Section 1, Section 2.
ButterflyLab is a MATLAB toolbox for fast evaluation of multidimensional Fourier integral operators for wave equations and a class of transforms in harmonic analysis, and fast solvers for high-frequency integral equations. Algorithms in this package are based on the comprementary low-rank structure of the matrix representations of these opertors and transforms. The butterfly algorithm or butterfly factorization is able to evaluate the matrix-vector multiplication with nearly linear operation and memory complexity. This MATLAB package is available at Codes.
Motivated by the extremely large amounts of data streaming from scientific research and daily life, recent years have witnessed exciting developments in the theory and practice of numerical linear algebra for obtaining instantaneous insight from massive datasets. This minisymposium introduces recent advances in both theory and high-performance algorithms for fast numerical linear algebra in data science, e.g. efficient matrix decomposition and applications, fast and scalable algorithms for optimization. These techniques significantly improve the efficiency of existing algorithms for information recovery, statistical learning, and machine learning. Section 1, Section 2.
Hierarchical structures and randomness have led to many efficient linear algebra subroutines that make large-scale problems feasible. This minisymposium introduces recent advances in high performance algorithms for fast linear algebra based on hierarchical structure and randomization techniques. These techniques result in linear or quasi-linear time methods for matrix operation, e.g., matrix multiplication, factorization, and inversion (or preconditioner). The fields of applications include fast solvers for PDEs, applied harmonic analysis, inverse problems, machine learning, etc. Section 1, Section 2.
This minisymposium introduces new advances in the mode decomposition problem with an emphasis on multidimensional data analysis. This problem aims at identifying and separating pre-assumed data patterns from their superposition. It has motivated new mathematical theory and numerical tools in adaptive and nonlinear time-frequency analysis, data-driven optimization based on variational principles and dictionary learning for sparsity, etc. This three-part minisymposium will mainly focus on real applications including seismic data analysis in geophysics, painting analysis in art investigation, and atomic crystal image analysis in materials science, respectively. Section 1,Section 2, Section 3.
SynCrystal is a MATLAB toolbox to analyze atomic crystal images. For a given atomic crystal image, it contains several tools to identify grain boundary, crystal orientation, elastic deformation. A few examples of synthetic and real atomic crystal images are provided to illustrate how to use these tools. A MATLAB package is available at https://github.com/SynCrystal/SynCrystal.
The study of structured matrices has led to many efficient linear algebra subroutines that make large-scale problems feasible. This minisymposium introduces recent advances in fast algorithms for computing data-sparse approximations of structured matrices based on randomization techniques, hierarchical matrix structure, FMM, etc. These data-sparse approximations give linear or quasi-linear time methods for eigendecomposition, matrix factorization, and multiplication with applications in fast solvers for high dimensional PDE’s, integral transforms (e.g., Fourier integral operators) and special function transforms. Link to minisymposium.
In mode decomposition problems, people process decompose signals that are superpositions of smoothly deformed plane waves. The wave shape of each component is a trigonometric function. In the general mode decomposition problem, each component has a general wave shape function that is corresponding to a certain physical meaning instead of a trigonomentric function making mode decomposition problem much more difficult. We addressed this problem in a recent paper 'Synchrosqueezed Wave Packet Transforms and Diffeomorphism Based Spectral Analysis for 1D General Mode Decompositions' in Applied and Computational Harmonic Analysis, 2014. A MATLAB package has been added in SynLab (https://github.com/HaizhaoYang/SynLab).