Applied/ACMS/absS23: Difference between revisions
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Abstract: We present a framework for simulating classical dynamical systems by finite-dimensional quantum system amenable to implementation on a quantum computer. Using ideas from kernel-based machine learning, the framework employs a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space (RKHS). Simultaneously, a mapping is employed to represent classical observables by quantum observables on the RKHS such that quantum mechanical expectation values are consistent with pointwise function evaluation. With this approach, quantum states and observables evolve under the Koopman operator of the dynamical system in a consistent manner with classical evolution. Moreover, the state of the quantum system can be projected onto a finite-rank density operator on a tensor product Hilbert space, enabling efficient implementation in a quantum circuit. We illustrate our approach with quantum circuit simulations of low-dimensional dynamical systems, as well as actual experiments on the IBM Quantum System One. | Abstract: We present a framework for simulating classical dynamical systems by finite-dimensional quantum system amenable to implementation on a quantum computer. Using ideas from kernel-based machine learning, the framework employs a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space (RKHS). Simultaneously, a mapping is employed to represent classical observables by quantum observables on the RKHS such that quantum mechanical expectation values are consistent with pointwise function evaluation. With this approach, quantum states and observables evolve under the Koopman operator of the dynamical system in a consistent manner with classical evolution. Moreover, the state of the quantum system can be projected onto a finite-rank density operator on a tensor product Hilbert space, enabling efficient implementation in a quantum circuit. We illustrate our approach with quantum circuit simulations of low-dimensional dynamical systems, as well as actual experiments on the IBM Quantum System One. | ||
=== | === Stephen Wright (UW) === | ||
Title: Optimization in theory and practice | Title: Optimization in theory and practice | ||
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Abstract: Organisms ranging from bacteria to reptiles can be found in granular beds which are often flooded with water and other matter. Depending on their size and strength, they may move entirely within the pore space or rearrange the material locally in search of food and shelter. We will discuss the dynamics of limbless worm Lumbriculus variegatus as a model to understand evolution-based strategies developed by organisms which routinely live and move through such disordered porous environments. The worms are shown to employ elongation-contraction and transverse undulatory strokes to propel themselves through a wide range of mediums. Our analysis in terms of the rheology of the medium shows that the dual strokes can be used by active intruders to move effectively from water through the loose fluidizable surface layers to the well-consolidated bed below. We will demonstrate corresponding motion of magnetoelastic robots depending on the frequency of their undulatory strokes and body elasticity. We will then examine worm foraging in the porous medium modeled as a series of chambers connected by narrow passages where steric interactions with confining walls lead to significant barriers for transport. Their escape time as they collide with the boundaries and locate passages between the chambers will be discussed in terms of a boundary-following random walk model. | Abstract: Organisms ranging from bacteria to reptiles can be found in granular beds which are often flooded with water and other matter. Depending on their size and strength, they may move entirely within the pore space or rearrange the material locally in search of food and shelter. We will discuss the dynamics of limbless worm Lumbriculus variegatus as a model to understand evolution-based strategies developed by organisms which routinely live and move through such disordered porous environments. The worms are shown to employ elongation-contraction and transverse undulatory strokes to propel themselves through a wide range of mediums. Our analysis in terms of the rheology of the medium shows that the dual strokes can be used by active intruders to move effectively from water through the loose fluidizable surface layers to the well-consolidated bed below. We will demonstrate corresponding motion of magnetoelastic robots depending on the frequency of their undulatory strokes and body elasticity. We will then examine worm foraging in the porous medium modeled as a series of chambers connected by narrow passages where steric interactions with confining walls lead to significant barriers for transport. Their escape time as they collide with the boundaries and locate passages between the chambers will be discussed in terms of a boundary-following random walk model. | ||
=== Mihai Anitescu (Argonne National Laboratory) === | === Mihai Anitescu (Univ of Chicago and Argonne National Laboratory) === | ||
Title: | Title: Exponential Decay of Sensitivity and Domain Decomposition with Overlap for Dynamic and Other Graph-Indexed Optimization | ||
Abstract: | Abstract: Many engineering control and optimization problems occupy an extensive area of space and time. These include production cost models in energy or the control of central plants. Model predictive control, one of the favorite approaches for physics-centered control, generally rely on direct numerical solvers, which tend to run out of memory for such problems. Approaches relying on domain decomposition are then key to fit in memory, but theory for such second-order methods tends to be lacking in multicomponent systems. | ||
To address this issue, we prove that certain classes of graph-indexed optimization (GIO) problems exhibit exponential decay of sensitivity with respect to perturbation in the data. GIOs include dynamic optimization (for the linear graph), or distributed, including network control (for the mesh/network-time product graph). This feature allows for very efficient approximation, solutions, or policies based on domain decomposition with overlap relative to centralized or monolithic approaches. In particular, we prove that the proper efficiency metric increases exponentially fast with the overlap size. Immediate consequences of such behavior are that distributed control policies with overlap approach the performance of centralized policies exponentially fast and that Schwarz-type algorithms exhibit, in addition to exceptional parallelism and reduced memory footprint per subproblem, a linear rate of convergence that tends exponentially fast to zero. | |||
=== Rupert Klein (FU Berlin) === | === Rupert Klein (FU Berlin) === | ||
'''Wasow Lecture''' | '''''Wasow Lecture''''' | ||
Title: Mathematics: A key to climate research | Title: Mathematics: A key to climate research | ||
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3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical formalization can help structuring discussions and bridging language barriers in interdisciplinary research. | 3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical formalization can help structuring discussions and bridging language barriers in interdisciplinary research. | ||
=== Romit Maulik (ANL/PSU) === | |||
Title: Developing predictive models for dynamical systems with scientific machine learning | |||
Abstract: The aim of this presentation is to provide an update on the development of physics-informed machine learning (ML) models for time series forecasting of complex dynamical systems. Specifically, the talk will focus on the use of recurrent neural network architectures combined with compression techniques to generate low-order models for forecasting high-dimensional dynamical systems. Additionally, we will explore the application of scalable neural architecture search to discover high-performing neural architectures and quantify uncertainty in predictions obtained from data-driven methods. The presentation will also discuss how the construction of these low-order models can be used to dramatically accelerate many-query applications such as data-assimilation, by virtue of their differentiable nature. To illustrate the versatility of our approach, we will provide examples from both canonical and real-world datasets. Finally, the presentation will address some of the limitations of the proposed approaches and suggest potential future work to address them. | |||
=== John Schotland (Yale University) === | |||
Title: Quantum optics in random media | |||
Abstract: Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients. | |||
Bio: John C. Schotland is Professor of Mathematics at Yale University. Previously, he was Professor of Mathematics and Professor of Physics at the University of Michigan, where he was also the founding director of the Michigan Center for Applied and Interdisciplinary Mathematics (MCAIM). He received the MD and PhD degrees from the University of Pennsylvania. | |||
=== Pedram Hassanzadeh (Rice University) === | |||
Title: Integrating the spectral analyses of neural networks and nonlinear physics for explainability, generalizability, and stability | |||
Abstract: In recent years, there has been substantial interest in using deep neural networks (NNs) to improve the modeling and prediction of complex, multiscale, nonlinear dynamical systems such as turbulent flows and Earth’s climate. In idealized settings, there has been some progress for a wide range of applications from data-driven spatio-temporal forecasting to long-term emulation to subgrid-scale modeling. However, to make these approaches practical and operational, i.e., scalable to real-world problems, a number of major questions and challenges need to be addressed. These include 1) instabilities and the emergence of unphysical behavior, e.g., due to how errors amplify through NNs, 2) learning in the small-data regime, 3) interpretability based on physics, and 4) out-of-distribution generalization (e.g., extrapolation to different parameters, forcings, and regimes) which is essential for applications to non-stationary systems such as a changing climate. While some progress has been made in addressing (1)-(4), the approaches have been often ad-hoc, as currently there is no rigorous framework to analyze deep NNs and develop systematic and general solutions to (1)-(4). In this talk, I will discuss some of the approaches to address (1)-(4). Then I will introduce a new framework that combines the spectral (Fourier) analyses of NNs and nonlinear physics, and leverages recent advances in theory and applications of deep learning, to move toward rigorous analysis of deep NNs for applications involving dynamical systems. I will use examples from subgrid-scale modeling of 2D turbulence and Rayleigh-Bernard turbulence and forecasting extreme weather to discuss these methods and ideas. | |||
Latest revision as of 01:40, 3 January 2024
ACMS Abstracts: Spring 2023
Paul Milewski (Bath)
Title: Embedded solitary internal waves
Abstract: The ocean and atmosphere are density stratified fluids. Stratified fluids with narrow regions of rapid density variation with respect to depth (pycnoclines) are often modelled as layered flows. In this talk we shall examine horizontally propagating internal waves within a three-layer fluid, with a focus on mode-2 waves which have oscillatory vertical structure. Mode-2 nonlinear waves (typically) occur within the linear spectrum of mode-1 waves (i.e. they travel at lower speeds than mode-1 waves), and are hence generically associated with an unphysical resonant mode-1 oscillatory tail. We will present evidence that these tail oscillations can be found to have zero amplitude, thus resulting in families of localised solutions (so called embedded solitary waves) in the Euler equations. This is the first example we know of embedded solitary waves in the Euler equations.
Nimish Pujara (UW)
Title: Flow and friction on a beach due to breaking waves
Abstract: As water waves approach a beach, they undergo dramatic transformations that have significant consequences for beach morphology. The most important transformations for the flow dynamics are that waves usually break before they reach the shoreline and that their height collapses when they do reach the shoreline. In this talk, we consider these processes and the subsequent flow that is driven up the beach. We present measurements of this flow in large-scale experiments with a focus on understanding the flow evolution in space and time, its friction with the beach surface, and its potential to transport large amounts of sediment. We demonstrate the link between wave-driven flow on a beach and canonical solutions to the shallow water equations, which allows us to describe the flow using reduced-parameter models. Using measurements of the wall shear stress, we also show that the importance of friction is confined to a narrow region within the flow at the interface between the wet and dry portions of the beach, and we present a simplified model that considers the dynamics of this region. Finally, we discuss a few extensions of this work that have applications to understanding sediment transport and the risk of coastal flooding.
Dimitris Giannakis (Dartmouth)
Title: Quantum information for simulation of classical dynamics
Abstract: We present a framework for simulating classical dynamical systems by finite-dimensional quantum system amenable to implementation on a quantum computer. Using ideas from kernel-based machine learning, the framework employs a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space (RKHS). Simultaneously, a mapping is employed to represent classical observables by quantum observables on the RKHS such that quantum mechanical expectation values are consistent with pointwise function evaluation. With this approach, quantum states and observables evolve under the Koopman operator of the dynamical system in a consistent manner with classical evolution. Moreover, the state of the quantum system can be projected onto a finite-rank density operator on a tensor product Hilbert space, enabling efficient implementation in a quantum circuit. We illustrate our approach with quantum circuit simulations of low-dimensional dynamical systems, as well as actual experiments on the IBM Quantum System One.
Stephen Wright (UW)
Title: Optimization in theory and practice
Abstract: Complexity analysis in optimization seeks upper bounds on the amount of work required to find approximate solutions of problems in a given class with a given algorithm, and also lower bounds, usually in the form of a worst-case example from a given problem class. The relationship between theoretical complexity bounds and practical performance of algorithms on “typical” problems varies widely across problem and algorithm classes. Over the years, research emphasis has switched between the theoretical and practical aspects of algorithm design and analysis. This talk surveys complexity analysis and its relationship to practice in optimization, with an emphasis on linear programming and convex and nonconvex nonlinear optimization, providing historical (and cultural) perspectives on research in these areas.
Angel Adames-Corraliza (UW)
Title: Theory and observations that slow tropical motions transport latent energy poleward
Abstract: Interactions between large-scale waves and the Hadley Cell are examined using a linear two-layer model on an $f$-plane. A linear meridional moisture gradient determines the strength of the idealized Hadley cell. The trade winds are in thermal wind balance with a weak temperature gradient (WTG). The domain is in WTG balance and wave solutions take the form of moisture modes. The westward propagation of the waves is largely due to moisture advection by the trade winds. Meridional moisture advection renders them unstable, i.e. they grow from ``moisture-vortex instability". The instability results in a poleward eddy moisture flux that flattens the mean meridional moisture gradient, thereby weakening the Hadley Cell. A Hadley Cell-moisture mode interaction is found that is reminiscent of quasi-geostrophic wave-mean flow interactions, except that wave activity is due to column moisture variance rather than potential vorticity variance. WTG balance reduces the Lorenz energy cycle to kinetic energy generation and conversions between the mean flow and the eddies. The conversion of zonal mean kinetic energy to eddy kinetic energy is due to the poleward eddy moisture flux and hence the tendency in wave activity. Data from ERA5 shows that tropical depression-like waves ---which were previously identified to behave like moisture modes that grow from moisture-vortex instability-- and flux moisture poleward. An analogy is proposed in which moisture modes are the tropical analog to midlatitude baroclinic waves. Moisture-vortex instability is analogous to baroclinic instability, stirring latent energy in the same way that baroclinic eddies stir sensible heat.
Ehud Yariv (Technion)
Title: Flows about superhydrophobic surfaces
Abstract: Superhydrophobic surfaces, formed by air entrapment within the cavities of hydrophobic solid substrates, offer a promising potential for hydrodynamic drag reduction. In several of the prototypical surface geometries the flows are two-dimensional, governed by Laplace’s equation in the longitudinal problem and the biharmonic equation in the transverse problem. Moreover, low-drag configurations are typically associated with singular limits. Thus, the analysis of liquid slippage past superhydrophobic surfaces naturally invites the use of both singular-perturbation methods and conformal-mapping techniques. I will discuss the combined application of these methodologies to several emerging problems in the field.
Arshad Kudrolli (Clark)
Title: Swimming and burrowing in sand and water
Abstract: Organisms ranging from bacteria to reptiles can be found in granular beds which are often flooded with water and other matter. Depending on their size and strength, they may move entirely within the pore space or rearrange the material locally in search of food and shelter. We will discuss the dynamics of limbless worm Lumbriculus variegatus as a model to understand evolution-based strategies developed by organisms which routinely live and move through such disordered porous environments. The worms are shown to employ elongation-contraction and transverse undulatory strokes to propel themselves through a wide range of mediums. Our analysis in terms of the rheology of the medium shows that the dual strokes can be used by active intruders to move effectively from water through the loose fluidizable surface layers to the well-consolidated bed below. We will demonstrate corresponding motion of magnetoelastic robots depending on the frequency of their undulatory strokes and body elasticity. We will then examine worm foraging in the porous medium modeled as a series of chambers connected by narrow passages where steric interactions with confining walls lead to significant barriers for transport. Their escape time as they collide with the boundaries and locate passages between the chambers will be discussed in terms of a boundary-following random walk model.
Mihai Anitescu (Univ of Chicago and Argonne National Laboratory)
Title: Exponential Decay of Sensitivity and Domain Decomposition with Overlap for Dynamic and Other Graph-Indexed Optimization
Abstract: Many engineering control and optimization problems occupy an extensive area of space and time. These include production cost models in energy or the control of central plants. Model predictive control, one of the favorite approaches for physics-centered control, generally rely on direct numerical solvers, which tend to run out of memory for such problems. Approaches relying on domain decomposition are then key to fit in memory, but theory for such second-order methods tends to be lacking in multicomponent systems.
To address this issue, we prove that certain classes of graph-indexed optimization (GIO) problems exhibit exponential decay of sensitivity with respect to perturbation in the data. GIOs include dynamic optimization (for the linear graph), or distributed, including network control (for the mesh/network-time product graph). This feature allows for very efficient approximation, solutions, or policies based on domain decomposition with overlap relative to centralized or monolithic approaches. In particular, we prove that the proper efficiency metric increases exponentially fast with the overlap size. Immediate consequences of such behavior are that distributed control policies with overlap approach the performance of centralized policies exponentially fast and that Schwarz-type algorithms exhibit, in addition to exceptional parallelism and reduced memory footprint per subproblem, a linear rate of convergence that tends exponentially fast to zero.
Rupert Klein (FU Berlin)
Wasow Lecture
Title: Mathematics: A key to climate research
Abstract: Mathematics in climate research is often thought to be mainly a provider of techniques for solving, e.g., the atmosphere and ocean flow equations. Three examples elucidate that its role is much broader and deeper:
1) Climate modelers often employ reduced forms of “the flow equations” for efficiency. Mathematical analysis helps assessing the regimes of validity of such models and defining conditions under which they can be solved robustly.
2) Climate is defined as “weather statistics”, and climate research investigates its change in time in our “single realization of Earth” with all its complexity. The required reliable notions of time dependent statistics for sparse data in high dimensions, however, remain to be established. Recent mathematical research offers advanced data analysis techniques that could be “game changing” in this respect.
3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical formalization can help structuring discussions and bridging language barriers in interdisciplinary research.
Romit Maulik (ANL/PSU)
Title: Developing predictive models for dynamical systems with scientific machine learning
Abstract: The aim of this presentation is to provide an update on the development of physics-informed machine learning (ML) models for time series forecasting of complex dynamical systems. Specifically, the talk will focus on the use of recurrent neural network architectures combined with compression techniques to generate low-order models for forecasting high-dimensional dynamical systems. Additionally, we will explore the application of scalable neural architecture search to discover high-performing neural architectures and quantify uncertainty in predictions obtained from data-driven methods. The presentation will also discuss how the construction of these low-order models can be used to dramatically accelerate many-query applications such as data-assimilation, by virtue of their differentiable nature. To illustrate the versatility of our approach, we will provide examples from both canonical and real-world datasets. Finally, the presentation will address some of the limitations of the proposed approaches and suggest potential future work to address them.
John Schotland (Yale University)
Title: Quantum optics in random media
Abstract: Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.
Bio: John C. Schotland is Professor of Mathematics at Yale University. Previously, he was Professor of Mathematics and Professor of Physics at the University of Michigan, where he was also the founding director of the Michigan Center for Applied and Interdisciplinary Mathematics (MCAIM). He received the MD and PhD degrees from the University of Pennsylvania.
Pedram Hassanzadeh (Rice University)
Title: Integrating the spectral analyses of neural networks and nonlinear physics for explainability, generalizability, and stability
Abstract: In recent years, there has been substantial interest in using deep neural networks (NNs) to improve the modeling and prediction of complex, multiscale, nonlinear dynamical systems such as turbulent flows and Earth’s climate. In idealized settings, there has been some progress for a wide range of applications from data-driven spatio-temporal forecasting to long-term emulation to subgrid-scale modeling. However, to make these approaches practical and operational, i.e., scalable to real-world problems, a number of major questions and challenges need to be addressed. These include 1) instabilities and the emergence of unphysical behavior, e.g., due to how errors amplify through NNs, 2) learning in the small-data regime, 3) interpretability based on physics, and 4) out-of-distribution generalization (e.g., extrapolation to different parameters, forcings, and regimes) which is essential for applications to non-stationary systems such as a changing climate. While some progress has been made in addressing (1)-(4), the approaches have been often ad-hoc, as currently there is no rigorous framework to analyze deep NNs and develop systematic and general solutions to (1)-(4). In this talk, I will discuss some of the approaches to address (1)-(4). Then I will introduce a new framework that combines the spectral (Fourier) analyses of NNs and nonlinear physics, and leverages recent advances in theory and applications of deep learning, to move toward rigorous analysis of deep NNs for applications involving dynamical systems. I will use examples from subgrid-scale modeling of 2D turbulence and Rayleigh-Bernard turbulence and forecasting extreme weather to discuss these methods and ideas.