Parameterization Method for Unstable Manifolds of Standing Waves on the Line

Barker, Blake; James, J. D. Mireles; Morgan, Jalen

doi:10.1137/19m128243x

Cited by 6 publications

(7 citation statements)

References 74 publications

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“…The potential of this feature is shown in many preceding works (e.g. [3,8,30,50,59]) for obtaining global feature of dynamical systems. We then extend the locally validated stable manifolds through the time integration of the time-reversal desingularized vector fields.…”

Section: Basic Methodologymentioning

confidence: 91%

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Lessard¹,

Matsue²,

Takayasu³

2021

Preprint

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In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial conditions are studied. Combining dynamical systems machinery (e.g. phase space compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g. rigorous integrators, parameterization method for invariant manifolds), these unstable blow-up solutions are obtained as trajectories on stable manifolds of hyperbolic (saddle) equilibria at infinity. In this process, important features are obtained: smooth dependence of blow-up times on initial points near blow-up, level set distribution of blow-up times, singular behavior of blow-up times on unstable blow-up solutions, organization of the phase space via separatrices (stable manifolds). In particular, we show that unstable blow-up solutions themselves, and solutions defined globally in time connected by those blow-up solutions can separate initial conditions into two regions where solution trajectories are either globally bounded or blow-up, no matter how the large initial points are.

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Section: Basic Methodologymentioning

confidence: 91%

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Lessard¹,

Matsue²,

Takayasu³

2021

Preprint

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show abstract

“…The sparsity challenges can be handled by incorporating generative adversarial networks (GANs) [123]. The method of manufactured learning can leverage both mathematical functions (e.g., Chebyshev, Dickens, wavelet families) or physical sets (e.g., proper orthogonal decomposition modes [45,193,281], Lagrangian coherent structures [133,144,301], exact coherent structures [39,92,314,346], terrain induced vortices [88,284,329], traveling waves [33,132,155], periodic or relative periodic orbits [72]) to train

𝒩

without requiring an expensive “truth" labeled data. It is also pivotal when we construct self‐evolving surrogate models on locally embedded structures.…”

Section: Hybrid Analysis and Modelingmentioning

confidence: 99%

Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution

2021

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Most modeling approaches lie in either of the two categories: physics‐based or data‐driven. Recently, a third approach which is a combination of these deterministic and statistical models is emerging for scientific applications. To leverage these developments, our aim in this perspective paper is centered around exploring numerous principle concepts to address the challenges of (i) trustworthiness and generalizability in developing data‐driven models to shed light on understanding the fundamental trade‐offs in their accuracy and efficiency and (ii) seamless integration of interface learning and multifidelity coupling approaches that transfer and represent information between different entities, particularly when different scales are governed by different physics, each operating on a different level of abstraction. Addressing these challenges could enable the revolution of digital twin technologies for scientific and engineering applications.

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“…The sparsity challenges can be handled by incorporating generative adversarial networks (GANs) [218] . The method of manufactured learning can leverage both mathematical functions (e.g., Chebyshev, Dickens, wavelet families) or physical sets (e.g., proper orthogonal decomposition modes [96,219,220] , Lagrangian coherent structures [221][222][223] , exact coherent structures [224][225][226][227] , terrain induced vortices [228][229][230] , traveling waves [231][232][233] , periodic or relative periodic orbits [234] ) to train  without requiring an expensive "truth" labeled data. It is also pivotal when we construct self-evolving surrogate models on locally embedded structures.…”

Section: Figurementioning

confidence: 99%

Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution

San

Rasheed

Kvamsdal

2021

Preprint

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Most modeling approaches lie in either of the two categories: physics-based or datadriven. Recently, a third approach which is a combination of these deterministic and statistical models is emerging for scientific applications. To leverage these developments, our aim in this perspective paper is centered around exploring numerous principle concepts to address the challenges of (i) trustworthiness and generalizability in developing data-driven models to shed light on understanding the fundamental trade-offs in their accuracy and efficiency, and (ii) seamless integration of interface learning and multifidelity coupling approaches that transfer and represent information between different entities, particularly when different scales are governed by different physics, each operating on a different level of abstraction. Addressing these challenges could enable the revolution of digital twin technologies for scientific and engineering applications.

show abstract

Parameterization Method for Unstable Manifolds of Standing Waves on the Line

Cited by 6 publications

References 74 publications

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution

Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution

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